finxpsuclem - Mathbox for ML

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Theorem finxpsuclem 36876

Description: Lemma for finxpsuc 36877. (Contributed by ML, 24-Oct-2020.)

**Hypothesis**
Ref	Expression
finxpsuclem.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

**Assertion**
Ref	Expression
finxpsuclem	⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Distinct variable groups: 𝑛,𝑁,𝑥 𝑈,𝑛,𝑥 Allowed substitution hints: 𝐹(𝑥,𝑛)

Proof of Theorem finxpsuclem Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7896	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2	1	adantr 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → suc 𝑁 ∈ ω)
3		1on 8499	. . . . . . . . . . . . 13 ⊢ 1_o ∈ On
4	3	onordi 6480	. . . . . . . . . . . 12 ⊢ Ord 1_o
5		nnord 7878	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
6		ordsseleq 6398	. . . . . . . . . . . 12 ⊢ ((Ord 1_o ∧ Ord 𝑁) → (1_o ⊆ 𝑁 ↔ (1_o ∈ 𝑁 ∨ 1_o = 𝑁)))
7	4, 5, 6	sylancr 586	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (1_o ⊆ 𝑁 ↔ (1_o ∈ 𝑁 ∨ 1_o = 𝑁)))
8	7	biimpa 476	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (1_o ∈ 𝑁 ∨ 1_o = 𝑁))
9		elelsuc 6442	. . . . . . . . . . . . 13 ⊢ (1_o ∈ 𝑁 → 1_o ∈ suc 𝑁)
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1_o ∈ 𝑁 → 1_o ∈ suc 𝑁))
11		sucidg 6450	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12		eleq1 2817	. . . . . . . . . . . . 13 ⊢ (1_o = 𝑁 → (1_o ∈ suc 𝑁 ↔ 𝑁 ∈ suc 𝑁))
13	11, 12	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1_o = 𝑁 → 1_o ∈ suc 𝑁))
14	10, 13	jaod 858	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1_o ∈ 𝑁 ∨ 1_o = 𝑁) → 1_o ∈ suc 𝑁))
15	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → ((1_o ∈ 𝑁 ∨ 1_o = 𝑁) → 1_o ∈ suc 𝑁))
16	8, 15	mpd 15	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → 1_o ∈ suc 𝑁)
17		finxpsuclem.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
18	17	finxpreclem6 36875	. . . . . . . . 9 ⊢ ((suc 𝑁 ∈ ω ∧ 1_o ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
19	2, 16, 18	syl2anc 583	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
20	19	sselda 3980	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
21	1	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22		df-2o 8488	. . . . . . . . . . . . . . 15 ⊢ 2_o = suc 1_o
23		ordsucsssuc 7826	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1_o ∧ Ord 𝑁) → (1_o ⊆ 𝑁 ↔ suc 1_o ⊆ suc 𝑁))
24	4, 5, 23	sylancr 586	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (1_o ⊆ 𝑁 ↔ suc 1_o ⊆ suc 𝑁))
25	24	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → suc 1_o ⊆ suc 𝑁)
26	22, 25	eqsstrid 4028	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → 2_o ⊆ suc 𝑁)
27	26	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2_o ⊆ suc 𝑁)
28		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
29	17	finxpreclem4 36873	. . . . . . . . . . . . 13 ⊢ (((suc 𝑁 ∈ ω ∧ 2_o ⊆ suc 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁))
30	21, 27, 28, 29	syl21anc 837	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁))
31		ordunisuc 7835	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑁 → ∪ suc 𝑁 = 𝑁)
32	5, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → ∪ suc 𝑁 = 𝑁)
33		opeq1 4874	. . . . . . . . . . . . . . . 16 ⊢ (∪ suc 𝑁 = 𝑁 → ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩ = ⟨𝑁, (1^st ‘𝑦)⟩)
34		rdgeq2 8433	. . . . . . . . . . . . . . . 16 ⊢ (⟨∪ suc 𝑁, (1^st ‘𝑦)⟩ = ⟨𝑁, (1^st ‘𝑦)⟩ → rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
35	33, 34	syl 17	. . . . . . . . . . . . . . 15 ⊢ (∪ suc 𝑁 = 𝑁 → rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
36	32, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
37	36, 32	fveq12d 6904	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
38	37	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
39	30, 38	eqtrd 2768	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
40	39	eqeq2d 2739	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
41	17	dffinxpf 36864	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
42	41	eqabri 2873	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
43	1	biantrurd 532	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
44	42, 43	bitr4id 290	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
45	44	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46		fvex 6910	. . . . . . . . . . . . 13 ⊢ (1^st ‘𝑦) ∈ V
47		opeq2 4875	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (1^st ‘𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1^st ‘𝑦)⟩)
48		rdgeq2 8433	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑁, 𝑧⟩ = ⟨𝑁, (1^st ‘𝑦)⟩ → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
49	47, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (1^st ‘𝑦) → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
50	49	fveq1d 6899	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1^st ‘𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
51	50	eqeq2d 2739	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1^st ‘𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
52	51	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1^st ‘𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))))
53	17	dffinxpf 36864	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
54	46, 52, 53	elab2 3671	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
55	54	baib 535	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
56	55	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
57	40, 45, 56	3bitr4d 311	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
58	57	biimpd 228	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
59	58	impancom 451	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (𝑦 ∈ (V × 𝑈) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
60	20, 59	mpd 15	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁))
61	60	ex 412	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
62	20	ex 412	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
63	61, 62	jcad 512	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
64	57	exbiri 810	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑦 ∈ (V × 𝑈) → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
65	64	impd 410	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
66	65	ancomsd 465	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
67	63, 66	impbid 211	. . 3 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68		elxp8 36850	. . 3 ⊢ (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
69	67, 68	bitr4di 289	. 2 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
70	69	eqrdv 2726	1 ⊢ ((𝑁 ∈ ω ∧ 1_o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 ∅c0 4323 ifcif 4529 ⟨cop 4635 ∪ cuni 4908 × cxp 5676 Ord word 6368 suc csuc 6371 ‘cfv 6548 ∈ cmpo 7422 ωcom 7870 1^st c1st 7991 reccrdg 8430 1_oc1o 8480 2_oc2o 8481 ↑↑cfinxp 36862

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 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740

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 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-en 8965 df-fin 8968 df-finxp 36863

This theorem is referenced by: finxpsuc 36877

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