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Theorem uniiccdif 25500

Description: A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)

**Hypothesis**
Ref	Expression
uniioombl.1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))

**Assertion**
Ref	Expression
uniiccdif	⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0))

Proof of Theorem uniiccdif Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun1 4168	. . 3 ⊢ ∪ ran ((,) ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
2		uniioombl.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3		ovolfcl 25388	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))))
4	2, 3	sylan 579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))))
5		rexr 11284	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥)) ∈ ℝ → (1^st ‘(𝐹‘𝑥)) ∈ ℝ^*)
6		rexr 11284	. . . . . . . 8 ⊢ ((2^nd ‘(𝐹‘𝑥)) ∈ ℝ → (2^nd ‘(𝐹‘𝑥)) ∈ ℝ^*)
7		id 22	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥)) → (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥)))
8		prunioo 13484	. . . . . . . 8 ⊢ (((1^st ‘(𝐹‘𝑥)) ∈ ℝ^* ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ^* ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))) → (((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ∪ {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))}) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
9	5, 6, 7, 8	syl3an 1158	. . . . . . 7 ⊢ (((1^st ‘(𝐹‘𝑥)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑥)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑥)) ≤ (2^nd ‘(𝐹‘𝑥))) → (((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ∪ {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))}) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
10	4, 9	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ∪ {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))}) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
11		fvco3 6991	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
12	2, 11	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
13	2	ffvelcdmda 7088	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
14	13	elin2d 4195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
15		1st2nd2 8026	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
16	14, 15	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
17	16	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
18		df-ov 7417	. . . . . . . . 9 ⊢ ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
19	17, 18	eqtr4di 2786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))
20	12, 19	eqtrd 2768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))))
21		df-pr 4627	. . . . . . . 8 ⊢ {((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)} = ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})
22		fvco3 6991	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
23	2, 22	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
24		fvco3 6991	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
25	2, 24	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
26	23, 25	preq12d 4741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → {((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)} = {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))})
27	21, 26	eqtr3id 2782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}) = {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))})
28	20, 27	uneq12d 4160	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})) = (((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ∪ {(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))}))
29		fvco3 6991	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
30	2, 29	sylan 579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
31	16	fveq2d 6895	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
32		df-ov 7417	. . . . . . . 8 ⊢ ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
33	31, 32	eqtr4di 2786	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
34	30, 33	eqtrd 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))))
35	10, 28, 34	3eqtr4rd 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})))
36	35	iuneq2dv 5015	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})))
37		iccf 13451	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
38		ffn 6716	. . . . . . 7 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ [,] Fn (ℝ^* × ℝ^*)
40		inss2 4225	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
41		rexpssxrxp 11283	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
42	40, 41	sstri 3987	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
43		fss 6733	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝐹:ℕ⟶(ℝ^ × ℝ^*))
44	2, 42, 43	sylancl 585	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ^* × ℝ^*))
45		fnfco 6756	. . . . . 6 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ([,] ∘ 𝐹) Fn ℕ)
46	39, 44, 45	sylancr 586	. . . . 5 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
47		fniunfv 7251	. . . . 5 ⊢ (([,] ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
48	46, 47	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
49		iunun 5090	. . . . 5 ⊢ ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})) = (∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}))
50		ioof 13450	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
51		ffn 6716	. . . . . . . . 9 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
52	50, 51	ax-mp 5	. . . . . . . 8 ⊢ (,) Fn (ℝ^* × ℝ^*)
53		fnfco 6756	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹) Fn ℕ)
54	52, 44, 53	sylancr 586	. . . . . . 7 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
55		fniunfv 7251	. . . . . . 7 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
56	54, 55	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
57		iunun 5090	. . . . . . 7 ⊢ ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}) = (∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)})
58		fo1st 8007	. . . . . . . . . . . . . 14 ⊢ 1^st :V–onto→V
59		fofn 6807	. . . . . . . . . . . . . 14 ⊢ (1^st :V–onto→V → 1^st Fn V)
60	58, 59	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1^st Fn V
61		ssv 4002	. . . . . . . . . . . . . 14 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ V
62		fss 6733	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ V) → 𝐹:ℕ⟶V)
63	2, 61, 62	sylancl 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶V)
64		fnfco 6756	. . . . . . . . . . . . 13 ⊢ ((1^st Fn V ∧ 𝐹:ℕ⟶V) → (1^st ∘ 𝐹) Fn ℕ)
65	60, 63, 64	sylancr 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ∘ 𝐹) Fn ℕ)
66		fnfun 6648	. . . . . . . . . . . 12 ⊢ ((1^st ∘ 𝐹) Fn ℕ → Fun (1^st ∘ 𝐹))
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (1^st ∘ 𝐹))
68		fndm 6651	. . . . . . . . . . . 12 ⊢ ((1^st ∘ 𝐹) Fn ℕ → dom (1^st ∘ 𝐹) = ℕ)
69		eqimss2 4037	. . . . . . . . . . . 12 ⊢ (dom (1^st ∘ 𝐹) = ℕ → ℕ ⊆ dom (1^st ∘ 𝐹))
70	65, 68, 69	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (1^st ∘ 𝐹))
71		dfimafn2 6956	. . . . . . . . . . 11 ⊢ ((Fun (1^st ∘ 𝐹) ∧ ℕ ⊆ dom (1^st ∘ 𝐹)) → ((1^st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)})
72	67, 70, 71	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)})
73		fnima 6679	. . . . . . . . . . 11 ⊢ ((1^st ∘ 𝐹) Fn ℕ → ((1^st ∘ 𝐹) “ ℕ) = ran (1^st ∘ 𝐹))
74	65, 73	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ∘ 𝐹) “ ℕ) = ran (1^st ∘ 𝐹))
75	72, 74	eqtr3d 2770	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} = ran (1^st ∘ 𝐹))
76		rnco2 6251	. . . . . . . . 9 ⊢ ran (1^st ∘ 𝐹) = (1^st “ ran 𝐹)
77	75, 76	eqtrdi 2784	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} = (1^st “ ran 𝐹))
78		fo2nd 8008	. . . . . . . . . . . . . 14 ⊢ 2^nd :V–onto→V
79		fofn 6807	. . . . . . . . . . . . . 14 ⊢ (2^nd :V–onto→V → 2^nd Fn V)
80	78, 79	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 2^nd Fn V
81		fnfco 6756	. . . . . . . . . . . . 13 ⊢ ((2^nd Fn V ∧ 𝐹:ℕ⟶V) → (2^nd ∘ 𝐹) Fn ℕ)
82	80, 63, 81	sylancr 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (2^nd ∘ 𝐹) Fn ℕ)
83		fnfun 6648	. . . . . . . . . . . 12 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → Fun (2^nd ∘ 𝐹))
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (2^nd ∘ 𝐹))
85		fndm 6651	. . . . . . . . . . . 12 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → dom (2^nd ∘ 𝐹) = ℕ)
86		eqimss2 4037	. . . . . . . . . . . 12 ⊢ (dom (2^nd ∘ 𝐹) = ℕ → ℕ ⊆ dom (2^nd ∘ 𝐹))
87	82, 85, 86	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ dom (2^nd ∘ 𝐹))
88		dfimafn2 6956	. . . . . . . . . . 11 ⊢ ((Fun (2^nd ∘ 𝐹) ∧ ℕ ⊆ dom (2^nd ∘ 𝐹)) → ((2^nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)})
89	84, 87, 88	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ∘ 𝐹) “ ℕ) = ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)})
90		fnima 6679	. . . . . . . . . . 11 ⊢ ((2^nd ∘ 𝐹) Fn ℕ → ((2^nd ∘ 𝐹) “ ℕ) = ran (2^nd ∘ 𝐹))
91	82, 90	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ∘ 𝐹) “ ℕ) = ran (2^nd ∘ 𝐹))
92	89, 91	eqtr3d 2770	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)} = ran (2^nd ∘ 𝐹))
93		rnco2 6251	. . . . . . . . 9 ⊢ ran (2^nd ∘ 𝐹) = (2^nd “ ran 𝐹)
94	92, 93	eqtrdi 2784	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)} = (2^nd “ ran 𝐹))
95	77, 94	uneq12d 4160	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ {((1^st ∘ 𝐹)‘𝑥)} ∪ ∪ 𝑥 ∈ ℕ {((2^nd ∘ 𝐹)‘𝑥)}) = ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
96	57, 95	eqtrid 2780	. . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)}) = ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
97	56, 96	uneq12d 4160	. . . . 5 ⊢ (𝜑 → (∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ∪ ∪ 𝑥 ∈ ℕ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})) = (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
98	49, 97	eqtrid 2780	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ ((((,) ∘ 𝐹)‘𝑥) ∪ ({((1^st ∘ 𝐹)‘𝑥)} ∪ {((2^nd ∘ 𝐹)‘𝑥)})) = (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
99	36, 48, 98	3eqtr3d 2776	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) = (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
100	1, 99	sseqtrrid 4031	. 2 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
101		ovolficcss 25391	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
102	2, 101	syl 17	. . . 4 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
103	102	ssdifssd 4138	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ)
104		omelon 9663	. . . . . . . . . . 11 ⊢ ω ∈ On
105		nnenom 13971	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
106	105	ensymi 9018	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
107		isnumi 9963	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
108	104, 106, 107	mp2an 691	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
109		fofun 6806	. . . . . . . . . . . . 13 ⊢ (1^st :V–onto→V → Fun 1^st )
110	58, 109	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1^st
111		ssv 4002	. . . . . . . . . . . . 13 ⊢ ran 𝐹 ⊆ V
112		fof 6805	. . . . . . . . . . . . . . 15 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
113	58, 112	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1^st :V⟶V
114	113	fdmi 6728	. . . . . . . . . . . . 13 ⊢ dom 1^st = V
115	111, 114	sseqtrri 4015	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 1^st
116		fores 6815	. . . . . . . . . . . 12 ⊢ ((Fun 1^st ∧ ran 𝐹 ⊆ dom 1^st ) → (1^st ↾ ran 𝐹):ran 𝐹–onto→(1^st “ ran 𝐹))
117	110, 115, 116	mp2an 691	. . . . . . . . . . 11 ⊢ (1^st ↾ ran 𝐹):ran 𝐹–onto→(1^st “ ran 𝐹)
118	2	ffnd 6717	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 Fn ℕ)
119		dffn4 6811	. . . . . . . . . . . 12 ⊢ (𝐹 Fn ℕ ↔ 𝐹:ℕ–onto→ran 𝐹)
120	118, 119	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ–onto→ran 𝐹)
121		foco 6819	. . . . . . . . . . 11 ⊢ (((1^st ↾ ran 𝐹):ran 𝐹–onto→(1^st “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((1^st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1^st “ ran 𝐹))
122	117, 120, 121	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → ((1^st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1^st “ ran 𝐹))
123		fodomnum 10074	. . . . . . . . . 10 ⊢ (ℕ ∈ dom card → (((1^st ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(1^st “ ran 𝐹) → (1^st “ ran 𝐹) ≼ ℕ))
124	108, 122, 123	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → (1^st “ ran 𝐹) ≼ ℕ)
125		domentr 9027	. . . . . . . . 9 ⊢ (((1^st “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (1^st “ ran 𝐹) ≼ ω)
126	124, 105, 125	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → (1^st “ ran 𝐹) ≼ ω)
127		fofun 6806	. . . . . . . . . . . . 13 ⊢ (2^nd :V–onto→V → Fun 2^nd )
128	78, 127	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2^nd
129		fof 6805	. . . . . . . . . . . . . . 15 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
130	78, 129	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2^nd :V⟶V
131	130	fdmi 6728	. . . . . . . . . . . . 13 ⊢ dom 2^nd = V
132	111, 131	sseqtrri 4015	. . . . . . . . . . . 12 ⊢ ran 𝐹 ⊆ dom 2^nd
133		fores 6815	. . . . . . . . . . . 12 ⊢ ((Fun 2^nd ∧ ran 𝐹 ⊆ dom 2^nd ) → (2^nd ↾ ran 𝐹):ran 𝐹–onto→(2^nd “ ran 𝐹))
134	128, 132, 133	mp2an 691	. . . . . . . . . . 11 ⊢ (2^nd ↾ ran 𝐹):ran 𝐹–onto→(2^nd “ ran 𝐹)
135		foco 6819	. . . . . . . . . . 11 ⊢ (((2^nd ↾ ran 𝐹):ran 𝐹–onto→(2^nd “ ran 𝐹) ∧ 𝐹:ℕ–onto→ran 𝐹) → ((2^nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2^nd “ ran 𝐹))
136	134, 120, 135	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2^nd “ ran 𝐹))
137		fodomnum 10074	. . . . . . . . . 10 ⊢ (ℕ ∈ dom card → (((2^nd ↾ ran 𝐹) ∘ 𝐹):ℕ–onto→(2^nd “ ran 𝐹) → (2^nd “ ran 𝐹) ≼ ℕ))
138	108, 136, 137	mpsyl 68	. . . . . . . . 9 ⊢ (𝜑 → (2^nd “ ran 𝐹) ≼ ℕ)
139		domentr 9027	. . . . . . . . 9 ⊢ (((2^nd “ ran 𝐹) ≼ ℕ ∧ ℕ ≈ ω) → (2^nd “ ran 𝐹) ≼ ω)
140	138, 105, 139	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → (2^nd “ ran 𝐹) ≼ ω)
141		unctb 10222	. . . . . . . 8 ⊢ (((1^st “ ran 𝐹) ≼ ω ∧ (2^nd “ ran 𝐹) ≼ ω) → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω)
142	126, 140, 141	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω)
143		ctex 8977	. . . . . . 7 ⊢ (((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∈ V)
144	142, 143	syl 17	. . . . . 6 ⊢ (𝜑 → ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∈ V)
145		ssid 4000	. . . . . . . 8 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
146	145, 99	sseqtrid 4030	. . . . . . 7 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
147		ssundif 4483	. . . . . . 7 ⊢ (∪ ran ([,] ∘ 𝐹) ⊆ (∪ ran ((,) ∘ 𝐹) ∪ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))) ↔ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
148	146, 147	sylib 217	. . . . . 6 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
149		ssdomg 9014	. . . . . 6 ⊢ (((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∈ V → ((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹))))
150	144, 148, 149	sylc 65	. . . . 5 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)))
151		domtr 9021	. . . . 5 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ∧ ((1^st “ ran 𝐹) ∪ (2^nd “ ran 𝐹)) ≼ ω) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
152	150, 142, 151	syl2anc 583	. . . 4 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω)
153		domentr 9027	. . . 4 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ω ∧ ω ≈ ℕ) → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
154	152, 106, 153	sylancl 585	. . 3 ⊢ (𝜑 → (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ)
155		ovolctb2 25414	. . 3 ⊢ (((∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ⊆ ℝ ∧ (∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹)) ≼ ℕ) → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
156	103, 154, 155	syl2anc 583	. 2 ⊢ (𝜑 → (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0)
157	100, 156	jca 511	1 ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ (vol*‘(∪ ran ([,] ∘ 𝐹) ∖ ∪ ran ((,) ∘ 𝐹))) = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ∖ cdif 3942 ∪ cun 3943 ∩ cin 3944 ⊆ wss 3945 𝒫 cpw 4598 {csn 4624 {cpr 4626 ⟨cop 4630 ∪ cuni 4903 ∪ ciun 4991 class class class wbr 5142 × cxp 5670 dom cdm 5672 ran crn 5673 ↾ cres 5674 “ cima 5675 ∘ ccom 5676 Oncon0 6363 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7414 ωcom 7864 1^st c1st 7985 2^nd c2nd 7986 ≈ cen 8954 ≼ cdom 8955 cardccrd 9952 ℝcr 11131 0cc0 11132 ℝ^*cxr 11271 ≤ cle 11273 ℕcn 12236 (,)cioo 13350 [,]cicc 13353 vol*covol 25384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-oi 9527 df-dju 9918 df-card 9956 df-acn 9959 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-xadd 13119 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659 df-xmet 21265 df-met 21266 df-ovol 25386

This theorem is referenced by: uniioombllem3 25507 uniioombllem4 25508 uniioombllem5 25509 uniiccmbl 25512

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