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Theorem ressress 17222

Description: Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)

**Assertion**
Ref	Expression
ressress	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))

**Proof of Theorem ressress**
Step	Hyp	Ref	Expression
1		simplr 768	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘𝑊) ⊆ 𝐴)
2		simpr1 1192	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝑊 ∈ V)
3		simpr2 1193	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		eqid 2728	. . . . . . . . . 10 ⊢ (𝑊 ↾_s 𝐴) = (𝑊 ↾_s 𝐴)
5		eqid 2728	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
6	4, 5	ressval2 17207	. . . . . . . . 9 ⊢ ((¬ (Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾_s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
7	1, 2, 3, 6	syl3anc 1369	. . . . . . . 8 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
8		inass 4215	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐴 ∩ (𝐵 ∩ (Base‘𝑊)))
9		in12 4216	. . . . . . . . . . 11 ⊢ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
10	8, 9	eqtri 2756	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
11	4, 5	ressbas 17208	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾_s 𝐴)))
12	3, 11	syl 17	. . . . . . . . . . 11 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾_s 𝐴)))
13	12	ineq2d 4208	. . . . . . . . . 10 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴))))
14	10, 13	eqtr2id 2781	. . . . . . . . 9 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
15	14	opeq2d 4876	. . . . . . . 8 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
16	7, 15	oveq12d 7432	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩) = ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
17		fvex 6904	. . . . . . . . 9 ⊢ (Base‘𝑊) ∈ V
18	17	inex2 5312	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) ∈ V
19		setsabs 17141	. . . . . . . 8 ⊢ ((𝑊 ∈ V ∧ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) ∈ V) → ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
20	2, 18, 19	sylancl 585	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
21	16, 20	eqtrd 2768	. . . . . 6 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
22		simpll 766	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵)
23		ovexd 7449	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) ∈ V)
24		simpr3 1194	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
25		eqid 2728	. . . . . . . 8 ⊢ ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = ((𝑊 ↾_s 𝐴) ↾_s 𝐵)
26		eqid 2728	. . . . . . . 8 ⊢ (Base‘(𝑊 ↾_s 𝐴)) = (Base‘(𝑊 ↾_s 𝐴))
27	25, 26	ressval2 17207	. . . . . . 7 ⊢ ((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ↾_s 𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = ((𝑊 ↾_s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩))
28	22, 23, 24, 27	syl3anc 1369	. . . . . 6 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = ((𝑊 ↾_s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾_s 𝐴)))⟩))
29		inss1 4224	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
30		sstr 3986	. . . . . . . . 9 ⊢ (((Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (Base‘𝑊) ⊆ 𝐴)
31	29, 30	mpan2 690	. . . . . . . 8 ⊢ ((Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) → (Base‘𝑊) ⊆ 𝐴)
32	1, 31	nsyl 140	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ¬ (Base‘𝑊) ⊆ (𝐴 ∩ 𝐵))
33		inex1g 5313	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
34	3, 33	syl 17	. . . . . . 7 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ 𝐵) ∈ V)
35		eqid 2728	. . . . . . . 8 ⊢ (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 ↾_s (𝐴 ∩ 𝐵))
36	35, 5	ressval2 17207	. . . . . . 7 ⊢ ((¬ (Base‘𝑊) ⊆ (𝐴 ∩ 𝐵) ∧ 𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
37	32, 2, 34, 36	syl3anc 1369	. . . . . 6 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
38	21, 28, 37	3eqtr4d 2778	. . . . 5 ⊢ (((¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ ¬ (Base‘𝑊) ⊆ 𝐴) ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
39	38	exp31 419	. . . 4 ⊢ (¬ (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 → (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))))
40		ovex 7447	. . . . . . . 8 ⊢ (𝑊 ↾_s 𝐴) ∈ V
41	25, 26	ressid2 17206	. . . . . . . 8 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ↾_s 𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s 𝐴))
42	40, 41	mp3an2 1446	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s 𝐴))
43	42	3ad2antr3 1188	. . . . . 6 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s 𝐴))
44		in32 4217	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)) = ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵)
45		simpr2 1193	. . . . . . . . . . . 12 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
46	45, 11	syl 17	. . . . . . . . . . 11 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾_s 𝐴)))
47		simpl 482	. . . . . . . . . . 11 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵)
48	46, 47	eqsstrd 4016	. . . . . . . . . 10 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵)
49		df-ss 3962	. . . . . . . . . 10 ⊢ ((𝐴 ∩ (Base‘𝑊)) ⊆ 𝐵 ↔ ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
50	48, 49	sylib 217	. . . . . . . . 9 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝐴 ∩ (Base‘𝑊)) ∩ 𝐵) = (𝐴 ∩ (Base‘𝑊)))
51	44, 50	eqtr2id 2781	. . . . . . . 8 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
52	51	oveq2d 7430	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐴 ∩ (Base‘𝑊))) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
53	5	ressinbas 17219	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑊 ↾_s 𝐴) = (𝑊 ↾_s (𝐴 ∩ (Base‘𝑊))))
54	45, 53	syl 17	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) = (𝑊 ↾_s (𝐴 ∩ (Base‘𝑊))))
55	5	ressinbas 17219	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
56	45, 33, 55	3syl 18	. . . . . . 7 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
57	52, 54, 56	3eqtr4d 2778	. . . . . 6 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
58	43, 57	eqtrd 2768	. . . . 5 ⊢ (((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
59	58	ex 412	. . . 4 ⊢ ((Base‘(𝑊 ↾_s 𝐴)) ⊆ 𝐵 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵))))
60	4, 5	ressid2 17206	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾_s 𝐴) = 𝑊)
61	60	3adant3r3 1182	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐴) = 𝑊)
62	61	oveq1d 7429	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s 𝐵))
63		inss2 4225	. . . . . . . . . . 11 ⊢ (𝐵 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
64		simpl 482	. . . . . . . . . . 11 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (Base‘𝑊) ⊆ 𝐴)
65	63, 64	sstrid 3989	. . . . . . . . . 10 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴)
66		sseqin2 4211	. . . . . . . . . 10 ⊢ ((𝐵 ∩ (Base‘𝑊)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
67	65, 66	sylib 217	. . . . . . . . 9 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴 ∩ (𝐵 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘𝑊)))
68	8, 67	eqtr2id 2781	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐵 ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
69	68	oveq2d 7430	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐵 ∩ (Base‘𝑊))) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
70		simpr3 1194	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
71	5	ressinbas 17219	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑌 → (𝑊 ↾_s 𝐵) = (𝑊 ↾_s (𝐵 ∩ (Base‘𝑊))))
72	70, 71	syl 17	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐵) = (𝑊 ↾_s (𝐵 ∩ (Base‘𝑊))))
73		simpr2 1193	. . . . . . . 8 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
74	73, 33, 55	3syl 18	. . . . . . 7 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = (𝑊 ↾_s ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))))
75	69, 72, 74	3eqtr4d 2778	. . . . . 6 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑊 ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
76	62, 75	eqtrd 2768	. . . . 5 ⊢ (((Base‘𝑊) ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
77	76	ex 412	. . . 4 ⊢ ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵))))
78	39, 59, 77	pm2.61ii 183	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
79	78	3expib 1120	. 2 ⊢ (𝑊 ∈ V → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵))))
80		ress0 17217	. . . 4 ⊢ (∅ ↾_s 𝐵) = ∅
81		reldmress 17204	. . . . . 6 ⊢ Rel dom ↾_s
82	81	ovprc1 7453	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾_s 𝐴) = ∅)
83	82	oveq1d 7429	. . . 4 ⊢ (¬ 𝑊 ∈ V → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (∅ ↾_s 𝐵))
84	81	ovprc1 7453	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾_s (𝐴 ∩ 𝐵)) = ∅)
85	80, 83, 84	3eqtr4a 2794	. . 3 ⊢ (¬ 𝑊 ∈ V → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))
86	85	a1d 25	. 2 ⊢ (¬ 𝑊 ∈ V → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵))))
87	79, 86	pm2.61i 182	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾_s 𝐴) ↾_s 𝐵) = (𝑊 ↾_s (𝐴 ∩ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ∩ cin 3944 ⊆ wss 3945 ∅c0 4318 ⟨cop 4630 ‘cfv 6542 (class class class)co 7414 sSet csts 17125 ndxcnx 17155 Basecbs 17173 ↾_s cress 17202

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-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-1cn 11190 ax-addcl 11192

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-ral 3058 df-rex 3067 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-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-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-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-nn 12237 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203

This theorem is referenced by: ressabs 17223 xrge00 32736 xrge0slmod 33054 fldexttr 33340 esumpfinvallem 33687 lmhmlnmsplit 42505

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