Theorem finxpsuclem 37892

Description: Lemma for finxpsuc 37893. (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 7871	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2	1	adantr 484	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 𝑁 ∈ ω)
3		1on 8451	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
4	3	onordi 6460	. . . . . . . . . . . 12 ⊢ Ord 1o
5		nnord 7855	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
6		ordsseleq 6376	. . . . . . . . . . . 12 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
7	4, 5, 6	sylancr 596	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
8	7	biimpa 480	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (1o ∈ 𝑁 ∨ 1o = 𝑁))
9		elelsuc 6422	. . . . . . . . . . . . 13 ⊢ (1o ∈ 𝑁 → 1o ∈ suc 𝑁)
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1o ∈ 𝑁 → 1o ∈ suc 𝑁))
11		sucidg 6430	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12		eleq1 2851	. . . . . . . . . . . . 13 ⊢ (1o = 𝑁 → (1o ∈ suc 𝑁 ↔ 𝑁 ∈ suc 𝑁))
13	11, 12	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1o = 𝑁 → 1o ∈ suc 𝑁))
14	10, 13	jaod 870	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1o ∈ 𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
15	14	adantr 484	. . . . . . . . . 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 37891	. . . . . . . . 9 ⊢ ((suc 𝑁 ∈ ω ∧ 1o ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
19	2, 16, 18	syl2anc 593	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
20	19	sselda 3937	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
21	1	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22		df-2o 8439	. . . . . . . . . . . . . . 15 ⊢ 2o = suc 1o
23		ordsucsssuc 7804	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
24	4, 5, 23	sylancr 596	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
25	24	biimpa 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 1o ⊆ suc 𝑁)
26	22, 25	eqsstrid 3975	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → 2o ⊆ suc 𝑁)
27	26	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2o ⊆ suc 𝑁)
28		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
29	17	finxpreclem4 37889	. . . . . . . . . . . . 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 848	. . . . . . . . . . . 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 7813	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑁 → ∪ suc 𝑁 = 𝑁)
32	5, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → ∪ suc 𝑁 = 𝑁)
33		opeq1 4832	. . . . . . . . . . . . . . . 16 ⊢ (∪ suc 𝑁 = 𝑁 → ⟨∪ suc 𝑁, (1st ‘𝑦)⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
34		rdgeq2 8384	. . . . . . . . . . . . . . . 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 6875	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → (rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
38	37	ad2antrr 736	. . . . . . . . . . . 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 2798	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
40	39	eqeq2d 2774	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
41	17	dffinxpf 37880	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
42	41	eqabri 2905	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
43	1	biantrurd 540	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
44	42, 43	bitr4id 292	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
45	44	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46		fvex 6881	. . . . . . . . . . . . 13 ⊢ (1st ‘𝑦) ∈ V
47		opeq2 4833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (1st ‘𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
48		rdgeq2 8384	. . . . . . . . . . . . . . . . 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 6870	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1st ‘𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
51	50	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1st ‘𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
52	51	anbi2d 639	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))))
53	17	dffinxpf 37880	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
54	46, 52, 53	elab2 3642	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
55	54	baib 543	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
56	55	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
57	40, 45, 56	3bitr4d 313	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
58	57	biimpd 231	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
59	58	impancom 455	. . . . . . 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 416	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
62	20	ex 416	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
63	61, 62	jcad 520	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
64	57	exbiri 820	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (V × 𝑈) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
65	64	impd 414	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1st ‘𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
66	65	ancomsd 469	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
67	63, 66	impbid 214	. . 3 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68		elxp8 37866	. . 3 ⊢ (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
69	67, 68	bitr4di 291	. 2 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
70	69	eqrdv 2761	1 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ifcif 4481  ⟨cop 4589  ∪ cuni 4866   × cxp 5646  Ord word 6346  suc csuc 6349  ‘cfv 6522   ∈ cmpo 7399  ωcom 7847  1^st c1st 7969  reccrdg 8381  1_oc1o 8431  2_oc2o 8432  ↑↑cfinxp 37878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-2o 8439  df-oadd 8442  df-en 8929  df-fin 8932  df-finxp 37879

This theorem is referenced by:  finxpsuc  37893