finxpsuclem - Mathbox for ML

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Theorem finxpsuclem 32410

Description: Lemma for finxpsuc 32411. (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 6978	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2	1	adantr 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → suc 𝑁 ∈ ω)
3		1on 7454	. . . . . . . . . . . . 13 ⊢ 1_𝑜 ∈ On
4	3	onordi 5749	. . . . . . . . . . . 12 ⊢ Ord 1_𝑜
5		nnord 6965	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
6		ordsseleq 5669	. . . . . . . . . . . 12 ⊢ ((Ord 1_𝑜 ∧ Ord 𝑁) → (1_𝑜 ⊆ 𝑁 ↔ (1_𝑜 ∈ 𝑁 ∨ 1_𝑜 = 𝑁)))
7	4, 5, 6	sylancr 694	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (1_𝑜 ⊆ 𝑁 ↔ (1_𝑜 ∈ 𝑁 ∨ 1_𝑜 = 𝑁)))
8	7	biimpa 500	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (1_𝑜 ∈ 𝑁 ∨ 1_𝑜 = 𝑁))
9		elelsuc 5714	. . . . . . . . . . . . 13 ⊢ (1_𝑜 ∈ 𝑁 → 1_𝑜 ∈ suc 𝑁)
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1_𝑜 ∈ 𝑁 → 1_𝑜 ∈ suc 𝑁))
11		sucidg 5720	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (1_𝑜 = 𝑁 → (1_𝑜 ∈ suc 𝑁 ↔ 𝑁 ∈ suc 𝑁))
13	11, 12	syl5ibrcom 236	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1_𝑜 = 𝑁 → 1_𝑜 ∈ suc 𝑁))
14	10, 13	jaod 394	. . . . . . . . . . 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 32409	. . . . . . . . 9 ⊢ ((suc 𝑁 ∈ ω ∧ 1_𝑜 ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
19	2, 16, 18	syl2anc 691	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
20	19	sselda 3568	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
21	1	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22		df-2o 7448	. . . . . . . . . . . . . . 15 ⊢ 2_𝑜 = suc 1_𝑜
23		ordsucsssuc 6915	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1_𝑜 ∧ Ord 𝑁) → (1_𝑜 ⊆ 𝑁 ↔ suc 1_𝑜 ⊆ suc 𝑁))
24	4, 5, 23	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (1_𝑜 ⊆ 𝑁 ↔ suc 1_𝑜 ⊆ suc 𝑁))
25	24	biimpa 500	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → suc 1_𝑜 ⊆ suc 𝑁)
26	22, 25	syl5eqss 3612	. . . . . . . . . . . . . 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 476	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
29	17	finxpreclem4 32407	. . . . . . . . . . . . 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 1317	. . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑁 → ∪ suc 𝑁 = 𝑁)
32	5, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → ∪ suc 𝑁 = 𝑁)
33		opeq1 4340	. . . . . . . . . . . . . . . 16 ⊢ (∪ suc 𝑁 = 𝑁 → ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩ = ⟨𝑁, (1^st ‘𝑦)⟩)
34		rdgeq2 7395	. . . . . . . . . . . . . . . 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 6109	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
38	37	ad2antrr 758	. . . . . . . . . . . 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 2644	. . . . . . . . . . 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 2620	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
41	1	biantrurd 528	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
42	17	dffinxpf 32398	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
43	42	abeq2i 2722	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
44	41, 43	syl6rbbr 278	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
45	44	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46		fvex 6113	. . . . . . . . . . . . 13 ⊢ (1^st ‘𝑦) ∈ V
47		opeq2 4341	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (1^st ‘𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1^st ‘𝑦)⟩)
48		rdgeq2 7395	. . . . . . . . . . . . . . . . 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 6105	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1^st ‘𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
51	50	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1^st ‘𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
52	51	anbi2d 736	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1^st ‘𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))))
53	17	dffinxpf 32398	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
54	46, 52, 53	elab2 3323	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
55	54	baib 942	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
56	55	ad2antrr 758	. . . . . . . . . 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 299	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
58	57	biimpd 218	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
59	58	impancom 455	. . . . . . 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 449	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
62	20	ex 449	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
63	61, 62	jcad 554	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
64	57	exbiri 650	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (V × 𝑈) → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
65	64	impd 446	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
66	65	ancomsd 469	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
67	63, 66	impbid 201	. . 3 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68		elxp8 32395	. . 3 ⊢ (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
69	67, 68	syl6bbr 277	. 2 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
70	69	eqrdv 2608	1 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ⟨cop 4131 ∪ cuni 4372 × cxp 5036 Ord word 5639 suc csuc 5642 ‘cfv 5804 ↦ cmpt2 6551 ωcom 6957 1^st c1st 7057 reccrdg 7392 1_𝑜c1o 7440 2_𝑜c2o 7441 ↑↑cfinxp 32396

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-finxp 32397

This theorem is referenced by: finxpsuc 32411

W3C validator