finxpsuclem - Mathbox for ML

	Mathbox for ML		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > finxpsuclem		Structured version Visualization version GIF version

Theorem finxpsuclem 33234

Description: Lemma for finxpsuc 33235. (Contributed by ML, 24-Oct-2020.)

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

**Assertion**
Ref	Expression
finxpsuclem	⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 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 7086	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2	1	adantr 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → suc 𝑁 ∈ ω)
3		1on 7567	. . . . . . . . . . . . 13 ⊢ 1_𝑜 ∈ On
4	3	onordi 5832	. . . . . . . . . . . 12 ⊢ Ord 1_𝑜
5		nnord 7073	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → Ord 𝑁)
6		ordsseleq 5752	. . . . . . . . . . . 12 ⊢ ((Ord 1_𝑜 ∧ Ord 𝑁) → (1_𝑜 ⊆ 𝑁 ↔ (1_𝑜 ∈ 𝑁 ∨ 1_𝑜 = 𝑁)))
7	4, 5, 6	sylancr 695	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (1_𝑜 ⊆ 𝑁 ↔ (1_𝑜 ∈ 𝑁 ∨ 1_𝑜 = 𝑁)))
8	7	biimpa 501	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (1_𝑜 ∈ 𝑁 ∨ 1_𝑜 = 𝑁))
9		elelsuc 5797	. . . . . . . . . . . . 13 ⊢ (1_𝑜 ∈ 𝑁 → 1_𝑜 ∈ suc 𝑁)
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1_𝑜 ∈ 𝑁 → 1_𝑜 ∈ suc 𝑁))
11		sucidg 5803	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → 𝑁 ∈ suc 𝑁)
12		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (1_𝑜 = 𝑁 → (1_𝑜 ∈ suc 𝑁 ↔ 𝑁 ∈ suc 𝑁))
13	11, 12	syl5ibrcom 237	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (1_𝑜 = 𝑁 → 1_𝑜 ∈ suc 𝑁))
14	10, 13	jaod 395	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1_𝑜 ∈ 𝑁 ∨ 1_𝑜 = 𝑁) → 1_𝑜 ∈ suc 𝑁))
15	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → ((1_𝑜 ∈ 𝑁 ∨ 1_𝑜 = 𝑁) → 1_𝑜 ∈ suc 𝑁))
16	8, 15	mpd 15	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → 1_𝑜 ∈ suc 𝑁)
17		finxpsuclem.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
18	17	finxpreclem6 33233	. . . . . . . . 9 ⊢ ((suc 𝑁 ∈ ω ∧ 1_𝑜 ∈ suc 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
19	2, 16, 18	syl2anc 693	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑈↑↑suc 𝑁) ⊆ (V × 𝑈))
20	19	sselda 3603	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → 𝑦 ∈ (V × 𝑈))
21	1	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → suc 𝑁 ∈ ω)
22		df-2o 7561	. . . . . . . . . . . . . . 15 ⊢ 2_𝑜 = suc 1_𝑜
23		ordsucsssuc 7023	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1_𝑜 ∧ Ord 𝑁) → (1_𝑜 ⊆ 𝑁 ↔ suc 1_𝑜 ⊆ suc 𝑁))
24	4, 5, 23	sylancr 695	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ω → (1_𝑜 ⊆ 𝑁 ↔ suc 1_𝑜 ⊆ suc 𝑁))
25	24	biimpa 501	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → suc 1_𝑜 ⊆ suc 𝑁)
26	22, 25	syl5eqss 3649	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → 2_𝑜 ⊆ suc 𝑁)
27	26	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 2_𝑜 ⊆ suc 𝑁)
28		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (V × 𝑈))
29	17	finxpreclem4 33231	. . . . . . . . . . . . 13 ⊢ (((suc 𝑁 ∈ ω ∧ 2_𝑜 ⊆ suc 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁))
30	21, 27, 28, 29	syl21anc 1325	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁))
31		ordunisuc 7032	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑁 → ∪ suc 𝑁 = 𝑁)
32	5, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ω → ∪ suc 𝑁 = 𝑁)
33		opeq1 4402	. . . . . . . . . . . . . . . 16 ⊢ (∪ suc 𝑁 = 𝑁 → ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩ = ⟨𝑁, (1^st ‘𝑦)⟩)
34		rdgeq2 7508	. . . . . . . . . . . . . . . 16 ⊢ (⟨∪ suc 𝑁, (1^st ‘𝑦)⟩ = ⟨𝑁, (1^st ‘𝑦)⟩ → rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
35	33, 34	syl 17	. . . . . . . . . . . . . . 15 ⊢ (∪ suc 𝑁 = 𝑁 → rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
36	32, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ω → rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
37	36, 32	fveq12d 6197	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
38	37	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨∪ suc 𝑁, (1^st ‘𝑦)⟩)‘∪ suc 𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
39	30, 38	eqtrd 2656	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
40	39	eqeq2d 2632	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
41	1	biantrurd 529	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → (∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))))
42	17	dffinxpf 33222	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑suc 𝑁) = {𝑦 ∣ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁))}
43	42	abeq2i 2735	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (suc 𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
44	41, 43	syl6rbbr 279	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
45	44	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ∅ = (rec(𝐹, ⟨suc 𝑁, 𝑦⟩)‘suc 𝑁)))
46		fvex 6201	. . . . . . . . . . . . 13 ⊢ (1^st ‘𝑦) ∈ V
47		opeq2 4403	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (1^st ‘𝑦) → ⟨𝑁, 𝑧⟩ = ⟨𝑁, (1^st ‘𝑦)⟩)
48		rdgeq2 7508	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑁, 𝑧⟩ = ⟨𝑁, (1^st ‘𝑦)⟩ → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
49	47, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (1^st ‘𝑦) → rec(𝐹, ⟨𝑁, 𝑧⟩) = rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩))
50	49	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1^st ‘𝑦) → (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))
51	50	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1^st ‘𝑦) → (∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
52	51	anbi2d 740	. . . . . . . . . . . . 13 ⊢ (𝑧 = (1^st ‘𝑦) → ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁))))
53	17	dffinxpf 33222	. . . . . . . . . . . . 13 ⊢ (𝑈↑↑𝑁) = {𝑧 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑧⟩)‘𝑁))}
54	46, 52, 53	elab2 3354	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
55	54	baib 944	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
56	55	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ↔ ∅ = (rec(𝐹, ⟨𝑁, (1^st ‘𝑦)⟩)‘𝑁)))
57	40, 45, 56	3bitr4d 300	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
58	57	biimpd 219	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
59	58	impancom 456	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (𝑦 ∈ (V × 𝑈) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
60	20, 59	mpd 15	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) ∧ 𝑦 ∈ (𝑈↑↑suc 𝑁)) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁))
61	60	ex 450	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)))
62	20	ex 450	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → 𝑦 ∈ (V × 𝑈)))
63	61, 62	jcad 555	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
64	57	exbiri 652	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (V × 𝑈) → ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) → 𝑦 ∈ (𝑈↑↑suc 𝑁))))
65	64	impd 447	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → ((𝑦 ∈ (V × 𝑈) ∧ (1^st ‘𝑦) ∈ (𝑈↑↑𝑁)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
66	65	ancomsd 470	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → 𝑦 ∈ (𝑈↑↑suc 𝑁)))
67	63, 66	impbid 202	. . 3 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈))))
68		elxp8 33219	. . 3 ⊢ (𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈) ↔ ((1^st ‘𝑦) ∈ (𝑈↑↑𝑁) ∧ 𝑦 ∈ (V × 𝑈)))
69	67, 68	syl6bbr 278	. 2 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑦 ∈ (𝑈↑↑suc 𝑁) ↔ 𝑦 ∈ ((𝑈↑↑𝑁) × 𝑈)))
70	69	eqrdv 2620	1 ⊢ ((𝑁 ∈ ω ∧ 1_𝑜 ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 ifcif 4086 ⟨cop 4183 ∪ cuni 4436 × cxp 5112 Ord word 5722 suc csuc 5725 ‘cfv 5888 ↦ cmpt2 6652 ωcom 7065 1^st c1st 7166 reccrdg 7505 1_𝑜c1o 7553 2_𝑜c2o 7554 ↑↑cfinxp 33220

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-finxp 33221

This theorem is referenced by: finxpsuc 33235

W3C validator