Theorem finxpsuclem 36581

Description: Lemma for finxpsuc 36582. (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 7883	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2	1	adantr 479	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 𝑁 ∈ ω)
3		1on 8480	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
4	3	onordi 6474	. . . . . . . . . . . 12 ⊢ Ord 1o
5		nnord 7865	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
6		ordsseleq 6392	. . . . . . . . . . . 12 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
7	4, 5, 6	sylancr 585	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
8	7	biimpa 475	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (1o ∈ 𝑁 ∨ 1o = 𝑁))
9		elelsuc 6436	. . . . . . . . . . . . 13 ⊢ (1o ∈ 𝑁 → 1o ∈ suc 𝑁)
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1o ∈ 𝑁 → 1o ∈ suc 𝑁))
11		sucidg 6444	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12		eleq1 2819	. . . . . . . . . . . . 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 855	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1o ∈ 𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
15	14	adantr 479	. . . . . . . . . 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 36580	. . . . . . . . 9 ⊢ ((suc 𝑁 ∈ ω ∧ 1o ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
19	2, 16, 18	syl2anc 582	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
20	19	sselda 3981	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
21	1	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22		df-2o 8469	. . . . . . . . . . . . . . 15 ⊢ 2o = suc 1o
23		ordsucsssuc 7813	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
24	4, 5, 23	sylancr 585	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
25	24	biimpa 475	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 1o ⊆ suc 𝑁)
26	22, 25	eqsstrid 4029	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → 2o ⊆ suc 𝑁)
27	26	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2o ⊆ suc 𝑁)
28		simpr 483	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
29	17	finxpreclem4 36578	. . . . . . . . . . . . 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 834	. . . . . . . . . . . 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 7822	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑁 → ∪ suc 𝑁 = 𝑁)
32	5, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → ∪ suc 𝑁 = 𝑁)
33		opeq1 4872	. . . . . . . . . . . . . . . 16 ⊢ (∪ suc 𝑁 = 𝑁 → ⟨∪ suc 𝑁, (1st ‘𝑦)⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
34		rdgeq2 8414	. . . . . . . . . . . . . . . 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 6897	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → (rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
38	37	ad2antrr 722	. . . . . . . . . . . 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 2770	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
40	39	eqeq2d 2741	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
41	17	dffinxpf 36569	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
42	41	eqabri 2875	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
43	1	biantrurd 531	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
44	42, 43	bitr4id 289	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
45	44	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46		fvex 6903	. . . . . . . . . . . . 13 ⊢ (1st ‘𝑦) ∈ V
47		opeq2 4873	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (1st ‘𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
48		rdgeq2 8414	. . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1st ‘𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
51	50	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1st ‘𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
52	51	anbi2d 627	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))))
53	17	dffinxpf 36569	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
54	46, 52, 53	elab2 3671	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
55	54	baib 534	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
56	55	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
57	40, 45, 56	3bitr4d 310	. . . . . . . . 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 450	. . . . . . 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 411	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
62	20	ex 411	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
63	61, 62	jcad 511	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
64	57	exbiri 807	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (V × 𝑈) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
65	64	impd 409	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1st ‘𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
66	65	ancomsd 464	. . . 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 36555	. . 3 ⊢ (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
69	67, 68	bitr4di 288	. 2 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
70	69	eqrdv 2728	1 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ⊆ wss 3947  ∅c0 4321  ifcif 4527  ⟨cop 4633  ∪ cuni 4907   × cxp 5673  Ord word 6362  suc csuc 6365  ‘cfv 6542   ∈ cmpo 7413  ωcom 7857  1^st c1st 7975  reccrdg 8411  1_oc1o 8461  2_oc2o 8462  ↑↑cfinxp 36567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-en 8942  df-fin 8945  df-finxp 36568

This theorem is referenced by:  finxpsuc  36582