Theorem finxpreclem5 32408

Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)

Hypothesis

Ref	Expression
finxpreclem5.1	⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))

Assertion

Ref	Expression
finxpreclem5	⊢ ((𝑛 ∈ ω ∧ 1𝑜 ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Distinct variable group:   𝑥,𝑛

Allowed substitution hints:   𝑈(𝑥,𝑛)   𝐹(𝑥,𝑛)

Proof of Theorem finxpreclem5

Step	Hyp	Ref	Expression
1		df-ov 6552	. . 3 ⊢ (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
3		0ex 4718	. . . . . . 7 ⊢ ∅ ∈ V
4		opex 4859	. . . . . . . 8 ⊢ ⟨∪ 𝑛, (1st ‘𝑥)⟩ ∈ V
5		opex 4859	. . . . . . . 8 ⊢ ⟨𝑛, 𝑥⟩ ∈ V
6	4, 5	ifex 4106	. . . . . . 7 ⊢ if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
7	3, 6	ifex 4106	. . . . . 6 ⊢ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
8		finxpreclem5.1	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
9	8	ovmpt4g 6681	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑥 ∈ V ∧ if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V) → (𝑛𝐹𝑥) = if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
10	2, 7, 9	mp3an23 1408	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
11	10	ad2antrr 758	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 1𝑜 ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
12		1on 7454	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ On
13	12	onirri 5751	. . . . . . . . . 10 ⊢ ¬ 1𝑜 ∈ 1𝑜
14		eleq2 2677	. . . . . . . . . 10 ⊢ (𝑛 = 1𝑜 → (1𝑜 ∈ 𝑛 ↔ 1𝑜 ∈ 1𝑜))
15	13, 14	mtbiri 316	. . . . . . . . 9 ⊢ (𝑛 = 1𝑜 → ¬ 1𝑜 ∈ 𝑛)
16	15	con2i 133	. . . . . . . 8 ⊢ (1𝑜 ∈ 𝑛 → ¬ 𝑛 = 1𝑜)
17	16	intnanrd 954	. . . . . . 7 ⊢ (1𝑜 ∈ 𝑛 → ¬ (𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈))
18	17	iffalsed 4047	. . . . . 6 ⊢ (1𝑜 ∈ 𝑛 → if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
19	18	adantl 481	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 1𝑜 ∈ 𝑛) → if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
20		iffalse 4045	. . . . 5 ⊢ (¬ 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
21	19, 20	sylan9eq 2664	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 1𝑜 ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → if((𝑛 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨𝑛, 𝑥⟩)
22	11, 21	eqtrd 2644	. . 3 ⊢ (((𝑛 ∈ ω ∧ 1𝑜 ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
23	1, 22	syl5eqr 2658	. 2 ⊢ (((𝑛 ∈ ω ∧ 1𝑜 ∈ 𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
24	23	ex 449	1 ⊢ ((𝑛 ∈ ω ∧ 1𝑜 ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))

Syntax hints:  ¬ wn 3   → 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-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447

This theorem is referenced by:  finxpreclem6  32409