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Theorem finxpreclem3 32406

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)

**Hypothesis**
Ref	Expression
finxpreclem3.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

**Assertion**
Ref	Expression
finxpreclem3	⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Distinct variable groups: 𝑛,𝑁,𝑥 𝑈,𝑛,𝑥 𝑛,𝑋,𝑥 Allowed substitution hints: 𝐹(𝑥,𝑛)

**Proof of Theorem finxpreclem3**
Step	Hyp	Ref	Expression
1		df-ov 6552	. 2 ⊢ (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
2		finxpreclem3.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
3	2	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))))
4		eqeq1 2614	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 = 1_𝑜 ↔ 𝑁 = 1_𝑜))
5		eleq1 2676	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
6	4, 5	bi2anan9 913	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈) ↔ (𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈)))
7		eleq1 2676	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8	7	adantl 481	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
9		unieq 4380	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ∪ 𝑛 = ∪ 𝑁)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ∪ 𝑛 = ∪ 𝑁)
11		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (1^st ‘𝑥) = (1^st ‘𝑋))
12	11	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → (1^st ‘𝑥) = (1^st ‘𝑋))
13	10, 12	opeq12d 4348	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨∪ 𝑛, (1^st ‘𝑥)⟩ = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
14		opeq12 4342	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
15	8, 13, 14	ifbieq12d 4063	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
16	6, 15	ifbieq2d 4061	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑥 = 𝑋) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if((𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)))
17		sssucid 5719	. . . . . . . . . . . . 13 ⊢ 1_𝑜 ⊆ suc 1_𝑜
18		df-2o 7448	. . . . . . . . . . . . 13 ⊢ 2_𝑜 = suc 1_𝑜
19	17, 18	sseqtr4i 3601	. . . . . . . . . . . 12 ⊢ 1_𝑜 ⊆ 2_𝑜
20		1on 7454	. . . . . . . . . . . . . 14 ⊢ 1_𝑜 ∈ On
21	18, 20	sucneqoni 32390	. . . . . . . . . . . . 13 ⊢ 2_𝑜 ≠ 1_𝑜
22	21	necomi 2836	. . . . . . . . . . . 12 ⊢ 1_𝑜 ≠ 2_𝑜
23		df-pss 3556	. . . . . . . . . . . 12 ⊢ (1_𝑜 ⊊ 2_𝑜 ↔ (1_𝑜 ⊆ 2_𝑜 ∧ 1_𝑜 ≠ 2_𝑜))
24	19, 22, 23	mpbir2an 957	. . . . . . . . . . 11 ⊢ 1_𝑜 ⊊ 2_𝑜
25		ssnpss 3672	. . . . . . . . . . 11 ⊢ (2_𝑜 ⊆ 1_𝑜 → ¬ 1_𝑜 ⊊ 2_𝑜)
26	24, 25	mt2 190	. . . . . . . . . 10 ⊢ ¬ 2_𝑜 ⊆ 1_𝑜
27		sseq2 3590	. . . . . . . . . 10 ⊢ (𝑁 = 1_𝑜 → (2_𝑜 ⊆ 𝑁 ↔ 2_𝑜 ⊆ 1_𝑜))
28	26, 27	mtbiri 316	. . . . . . . . 9 ⊢ (𝑁 = 1_𝑜 → ¬ 2_𝑜 ⊆ 𝑁)
29	28	con2i 133	. . . . . . . 8 ⊢ (2_𝑜 ⊆ 𝑁 → ¬ 𝑁 = 1_𝑜)
30	29	intnanrd 954	. . . . . . 7 ⊢ (2_𝑜 ⊆ 𝑁 → ¬ (𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈))
31	30	iffalsed 4047	. . . . . 6 ⊢ (2_𝑜 ⊆ 𝑁 → if((𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩))
32		iftrue 4042	. . . . . 6 ⊢ (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
33	31, 32	sylan9eq 2664	. . . . 5 ⊢ ((2_𝑜 ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1_𝑜 ∧ 𝑋 ∈ 𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨∪ 𝑁, (1^st ‘𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
34	16, 33	sylan9eqr 2666	. . . 4 ⊢ (((2_𝑜 ⊆ 𝑁 ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
35	34	adantlll 750	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
36		simpll 786	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
37		elex 3185	. . . 4 ⊢ (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
38	37	adantl 481	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
39		opex 4859	. . . 4 ⊢ ⟨∪ 𝑁, (1^st ‘𝑋)⟩ ∈ V
40	39	a1i 11	. . 3 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^st ‘𝑋)⟩ ∈ V)
41	3, 35, 36, 38, 40	ovmpt2d 6686	. 2 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨∪ 𝑁, (1^st ‘𝑋)⟩)
42	1, 41	syl5reqr 2659	1 ⊢ (((𝑁 ∈ ω ∧ 2_𝑜 ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ⊊ wpss 3541 ∅c0 3874 ifcif 4036 ⟨cop 4131 ∪ cuni 4372 × cxp 5036 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 1^st c1st 7057 1_𝑜c1o 7440 2_𝑜c2o 7441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1o 7447 df-2o 7448

This theorem is referenced by: finxpreclem4 32407

W3C validator