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Theorem finxpreclem1 32402

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)

**Assertion**
Ref	Expression
finxpreclem1	⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_𝑜, 𝑋⟩))

Distinct variable groups: 𝑈,𝑛,𝑥 𝑛,𝑋,𝑥

**Proof of Theorem finxpreclem1**
Step	Hyp	Ref	Expression
1		df-ov 6552	. 2 ⊢ (1_𝑜(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_𝑜, 𝑋⟩)
2		eqidd 2611	. . 3 ⊢ (𝑋 ∈ 𝑈 → (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))))
3		eleq1a 2683	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → 𝑥 ∈ 𝑈))
4	3	anim2d 587	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → ((𝑛 = 1_𝑜 ∧ 𝑥 = 𝑋) → (𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈)))
5		iftrue 4042	. . . . 5 ⊢ ((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅)
6	4, 5	syl6 34	. . . 4 ⊢ (𝑋 ∈ 𝑈 → ((𝑛 = 1_𝑜 ∧ 𝑥 = 𝑋) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅))
7	6	imp 444	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ (𝑛 = 1_𝑜 ∧ 𝑥 = 𝑋)) → if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅)
8		1onn 7606	. . . 4 ⊢ 1_𝑜 ∈ ω
9	8	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑈 → 1_𝑜 ∈ ω)
10		elex 3185	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V)
11		0ex 4718	. . . 4 ⊢ ∅ ∈ V
12	11	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑈 → ∅ ∈ V)
13	2, 7, 9, 10, 12	ovmpt2d 6686	. 2 ⊢ (𝑋 ∈ 𝑈 → (1_𝑜(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ∅)
14	1, 13	syl5reqr 2659	1 ⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_𝑜, 𝑋⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ifcif 4036 ⟨cop 4131 ∪ cuni 4372 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 1^st c1st 7057 1_𝑜c1o 7440

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-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1o 7447

This theorem is referenced by: finxp1o 32405

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