Theorem finxpsuclem 38063

Description: Lemma for finxpsuc 38064. (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 7882	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2	1	adantr 485	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → suc 𝑁 ∈ ω)
3		1on 8462	. . . . . . . . . . . . 13 ⊢ 1o ∈ On
4	3	onordi 6474	. . . . . . . . . . . 12 ⊢ Ord 1o
5		nnord 7866	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
6		ordsseleq 6390	. . . . . . . . . . . 12 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
7	4, 5, 6	sylancr 598	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ (1o ∈ 𝑁 ∨ 1o = 𝑁)))
8	7	biimpa 481	. . . . . . . . . 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 2851	. . . . . . . . . . . . 13 ⊢ (1o = 𝑁 → (1o ∈ suc 𝑁 ↔ 𝑁 ∈ suc 𝑁))
13	11, 12	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1o = 𝑁 → 1o ∈ suc 𝑁))
14	10, 13	jaod 872	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1o ∈ 𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
15	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → ((1o ∈ 𝑁 ∨ 1o = 𝑁) → 1o ∈ suc 𝑁))
16	8, 15	mpd 16	. . . . . . . . 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 38062	. . . . . . . . 9 ⊢ ((suc 𝑁 ∈ ω ∧ 1o ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
19	2, 16, 18	syl2anc 595	. . . . . . . 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 738	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22		df-2o 8450	. . . . . . . . . . . . . . 15 ⊢ 2o = suc 1o
23		ordsucsssuc 7815	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1o ∧ Ord 𝑁) → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
24	4, 5, 23	sylancr 598	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ suc 1o ⊆ suc 𝑁))
25	24	biimpa 481	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2o ⊆ suc 𝑁)
28		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
29	17	finxpreclem4 38060	. . . . . . . . . . . . 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 850	. . . . . . . . . . . 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 7824	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑁 → ∪ suc 𝑁 = 𝑁)
32	5, 31	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → ∪ suc 𝑁 = 𝑁)
33		opeq1 4838	. . . . . . . . . . . . . . . 16 ⊢ (∪ suc 𝑁 = 𝑁 → ⟨∪ suc 𝑁, (1st ‘𝑦)⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
34		rdgeq2 8395	. . . . . . . . . . . . . . . 16 ⊢ (⟨∪ suc 𝑁, (1st ‘𝑦)⟩ = ⟨𝑁, (1st ‘𝑦)⟩ → rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
35	33, 34	syl 18	. . . . . . . . . . . . . . 15 ⊢ (∪ suc 𝑁 = 𝑁 → rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
36	32, 35	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
37	36, 32	fveq12d 6888	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → (rec(𝐹, ⟨∪ suc 𝑁, (1st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))
38	37	ad2antrr 738	. . . . . . . . . . . 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 38051	. . . . . . . . . . . . 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 541	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
44	42, 43	bitr4id 293	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
45	44	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46		fvex 6894	. . . . . . . . . . . . 13 ⊢ (1st ‘𝑦) ∈ V
47		opeq2 4839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (1st ‘𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st ‘𝑦)⟩)
48		rdgeq2 8395	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑁, 𝑧⟩ = ⟨𝑁, (1st ‘𝑦)⟩ → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
49	47, 48	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (1st ‘𝑦) → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩))
50	49	fveq1d 6883	. . . . . . . . . . . . . . 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 641	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1st ‘𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁))))
53	17	dffinxpf 38051	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
54	46, 52, 53	elab2 3641	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
55	54	baib 544	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
56	55	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1st ‘𝑦)⟩)‘𝑁)))
57	40, 45, 56	3bitr4d 314	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
58	57	biimpd 232	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
59	58	impancom 456	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (𝑦 ∈ (V × 𝑈) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
60	20, 59	mpd 16	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁))
61	60	ex 417	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1st ‘𝑦) ∈ (𝑈↑↑𝑁)))
62	20	ex 417	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
63	61, 62	jcad 521	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
64	57	exbiri 822	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (V × 𝑈) → ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
65	64	impd 415	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1st ‘𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
66	65	ancomsd 470	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
67	63, 66	impbid 215	. . 3 ⊢ ((𝑁 ∈ ω ∧ 1o ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68		elxp8 38037	. . 3 ⊢ (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
69	67, 68	bitr4di 292	. 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 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ifcif 4487  ⟨cop 4595  ∪ cuni 4872   × cxp 5659  Ord word 6359  suc csuc 6362  ‘cfv 6536   ∈ cmpo 7412  ωcom 7858  1^st c1st 7980  reccrdg 8392  1_oc1o 8442  2_oc2o 8443  ↑↑cfinxp 38049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-en 8940  df-fin 8943  df-finxp 38050

This theorem is referenced by:  finxpsuc  38064