Theorem finxpsuclem 35722

Description: Lemma for finxpsuc 35723. (Contributed by ML, 24-Oct-2020.)

Hypothesis

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

Assertion

Ref	Expression
finxpsuclem	⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 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 7810	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2	1	adantr 482	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 𝑁 ∈ ω)
3		1on 8384	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
4	3	onordi 6416	. . . . . . . . . . . 12 ⊢ Ord 1o
5		nnord 7793	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
6		ordsseleq 6336	. . . . . . . . . . . 12 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
7	4, 5, 6	sylancr 588	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
8	7	biimpa 478	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (1o ∈ 𝑁 ∨ 1o = 𝑁))
9		elelsuc 6379	. . . . . . . . . . . . 13 ⊢ (1o ∈ 𝑁 → 1o ∈ suc 𝑁)
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1o ∈ 𝑁 → 1o ∈ suc 𝑁))
11		sucidg 6387	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (1o = 𝑁 → (1o ∈ suc 𝑁 ↔ 𝑁 ∈ suc 𝑁))
13	11, 12	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1o = 𝑁 → 1o ∈ suc 𝑁))
14	10, 13	jaod 857	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1o ∈ 𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
15	14	adantr 482	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → ((1o ∈ 𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
16	8, 15	mpd 15	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → 1o ∈ suc 𝑁)
17		finxpsuclem.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
18	17	finxpreclem6 35721	. . . . . . . . 9 ⊢ ((suc 𝑁 ∈ ω ∧ 1o ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
19	2, 16, 18	syl2anc 585	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
20	19	sselda 3936	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
21	1	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22		df-2o 8373	. . . . . . . . . . . . . . 15 ⊢ 2o = suc 1o
23		ordsucsssuc 7741	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
24	4, 5, 23	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
25	24	biimpa 478	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 1o ⊆ suc 𝑁)
26	22, 25	eqsstrid 3984	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → 2o ⊆ suc 𝑁)
27	26	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2o ⊆ suc 𝑁)
28		simpr 486	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
29	17	finxpreclem4 35719	. . . . . . . . . . . . 13 ⊢ (((suc 𝑁 ∈ ω ∧ 2o ⊆ suc 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩)‘∪ suc 𝑁))
30	21, 27, 28, 29	syl21anc 836	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩)‘∪ suc 𝑁))
31		ordunisuc 7750	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑁 → ∪ suc 𝑁 = 𝑁)
32	5, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → ∪ suc 𝑁 = 𝑁)
33		opeq1 4822	. . . . . . . . . . . . . . . 16 ⊢ (∪ suc 𝑁 = 𝑁 → ⟨∪ suc 𝑁, (1st ‘𝑦)⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
34		rdgeq2 8318	. . . . . . . . . . . . . . . 16 ⊢ (⟨∪ suc 𝑁, (1st ‘𝑦)⟩ = ⟨𝑁, (1st ‘𝑦)⟩ → rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
35	33, 34	syl 17	. . . . . . . . . . . . . . 15 ⊢ (∪ suc 𝑁 = 𝑁 → rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
36	32, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
37	36, 32	fveq12d 6837	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → (rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
38	37	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
39	30, 38	eqtrd 2777	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
40	39	eqeq2d 2748	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
41	17	dffinxpf 35710	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
42	41	abeq2i 2874	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
43	1	biantrurd 534	. . . . . . . . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46		fvex 6843	. . . . . . . . . . . . 13 ⊢ (1st ‘𝑦) ∈ V
47		opeq2 4823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (1st ‘𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
48		rdgeq2 8318	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st ‘𝑦)⟩ → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
49	47, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (1st ‘𝑦) → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
50	49	fveq1d 6832	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1st ‘𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
51	50	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1st ‘𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
52	51	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))))
53	17	dffinxpf 35710	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
54	46, 52, 53	elab2 3627	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
55	54	baib 537	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
56	55	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
57	40, 45, 56	3bitr4d 311	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
58	57	biimpd 228	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
59	58	impancom 453	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (𝑦 ∈ (V × 𝑈) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
60	20, 59	mpd 15	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁))
61	60	ex 414	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
62	20	ex 414	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
63	61, 62	jcad 514	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
64	57	exbiri 809	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (V × 𝑈) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
65	64	impd 412	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1st ‘𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
66	65	ancomsd 467	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
67	63, 66	impbid 211	. . 3 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68		elxp8 35696	. . 3 ⊢ (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
69	67, 68	bitr4di 289	. 2 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
70	69	eqrdv 2735	1 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3442   ⊆ wss 3902  ∅c0 4274  ifcif 4478  ⟨cop 4584  ∪ cuni 4857   × cxp 5623  Ord word 6306  suc csuc 6309  ‘cfv 6484   ∈ cmpo 7344  ωcom 7785  1^st c1st 7902  reccrdg 8315  1_oc1o 8365  2_oc2o 8366  ↑↑cfinxp 35708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-1st 7904  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-2o 8373  df-oadd 8376  df-en 8810  df-fin 8813  df-finxp 35709

This theorem is referenced by:  finxpsuc  35723