Theorem finxpsuclem 35568

Description: Lemma for finxpsuc 35569. (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 7737	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2	1	adantr 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 𝑁 ∈ ω)
3		1on 8309	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
4	3	onordi 6371	. . . . . . . . . . . 12 ⊢ Ord 1o
5		nnord 7720	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
6		ordsseleq 6295	. . . . . . . . . . . 12 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
7	4, 5, 6	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
8	7	biimpa 477	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (1o ∈ 𝑁 ∨ 1o = 𝑁))
9		elelsuc 6338	. . . . . . . . . . . . 13 ⊢ (1o ∈ 𝑁 → 1o ∈ suc 𝑁)
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1o ∈ 𝑁 → 1o ∈ suc 𝑁))
11		sucidg 6344	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (1o = 𝑁 → (1o ∈ suc 𝑁 ↔ 𝑁 ∈ suc 𝑁))
13	11, 12	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1o = 𝑁 → 1o ∈ suc 𝑁))
14	10, 13	jaod 856	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1o ∈ 𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
15	14	adantr 481	. . . . . . . . . 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 35567	. . . . . . . . 9 ⊢ ((suc 𝑁 ∈ ω ∧ 1o ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
19	2, 16, 18	syl2anc 584	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
20	19	sselda 3921	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
21	1	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22		df-2o 8298	. . . . . . . . . . . . . . 15 ⊢ 2o = suc 1o
23		ordsucsssuc 7670	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
24	4, 5, 23	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
25	24	biimpa 477	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 1o ⊆ suc 𝑁)
26	22, 25	eqsstrid 3969	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → 2o ⊆ suc 𝑁)
27	26	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2o ⊆ suc 𝑁)
28		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
29	17	finxpreclem4 35565	. . . . . . . . . . . . 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 835	. . . . . . . . . . . 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 7679	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑁 → ∪ suc 𝑁 = 𝑁)
32	5, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → ∪ suc 𝑁 = 𝑁)
33		opeq1 4804	. . . . . . . . . . . . . . . 16 ⊢ (∪ suc 𝑁 = 𝑁 → ⟨∪ suc 𝑁, (1st ‘𝑦)⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
34		rdgeq2 8243	. . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → (rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
38	37	ad2antrr 723	. . . . . . . . . . . 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 2778	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
40	39	eqeq2d 2749	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
41	17	dffinxpf 35556	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
42	41	abeq2i 2875	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
43	1	biantrurd 533	. . . . . . . . . . . 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 723	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46		fvex 6787	. . . . . . . . . . . . 13 ⊢ (1st ‘𝑦) ∈ V
47		opeq2 4805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (1st ‘𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
48		rdgeq2 8243	. . . . . . . . . . . . . . . . 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 6776	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1st ‘𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
51	50	eqeq2d 2749	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1st ‘𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
52	51	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))))
53	17	dffinxpf 35556	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
54	46, 52, 53	elab2 3613	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
55	54	baib 536	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
56	55	ad2antrr 723	. . . . . . . . . 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 452	. . . . . . 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 413	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
62	20	ex 413	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
63	61, 62	jcad 513	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
64	57	exbiri 808	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (V × 𝑈) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
65	64	impd 411	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1st ‘𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
66	65	ancomsd 466	. . . 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 35542	. . 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 2736	1 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  ifcif 4459  ⟨cop 4567  ∪ cuni 4839   × cxp 5587  Ord word 6265  suc csuc 6268  ‘cfv 6433   ∈ cmpo 7277  ωcom 7712  1^st c1st 7829  reccrdg 8240  1_oc1o 8290  2_oc2o 8291  ↑↑cfinxp 35554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-en 8734  df-fin 8737  df-finxp 35555

This theorem is referenced by:  finxpsuc  35569