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Theorem finxpreclem1 33226

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 6653	. 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 2623	. . 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 2696	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → 𝑥 ∈ 𝑈))
4	3	anim2d 589	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → ((𝑛 = 1_𝑜 ∧ 𝑥 = 𝑋) → (𝑛 = 1_𝑜 ∧ 𝑥 ∈ 𝑈)))
5		iftrue 4092	. . . . 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 35	. . . 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 445	. . 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 7719	. . . 4 ⊢ 1_𝑜 ∈ ω
9	8	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑈 → 1_𝑜 ∈ ω)
10		elex 3212	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V)
11		0ex 4790	. . . 4 ⊢ ∅ ∈ V
12	11	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑈 → ∅ ∈ V)
13	2, 7, 9, 10, 12	ovmpt2d 6788	. 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 2671	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 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ifcif 4086 ⟨cop 4183 ∪ cuni 4436 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ωcom 7065 1^st c1st 7166 1_𝑜c1o 7553

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-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-rab 2921 df-v 3202 df-sbc 3436 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-br 4654 df-opab 4713 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-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1o 7560

This theorem is referenced by: finxp1o 33229

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