Theorem finxpreclem1 37390

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

Assertion

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

Distinct variable groups:   𝑈,𝑛,𝑥   𝑛,𝑋,𝑥

Proof of Theorem finxpreclem1

Step	Hyp	Ref	Expression
1		eqidd 2738	. . 3 ⊢ (𝑋 ∈ 𝑈 → (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))))
2		eleq1a 2836	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → 𝑥 ∈ 𝑈))
3	2	anim2d 612	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → ((𝑛 = 1o ∧ 𝑥 = 𝑋) → (𝑛 = 1o ∧ 𝑥 ∈ 𝑈)))
4		iftrue 4531	. . . . 5 ⊢ ((𝑛 = 1o ∧ 𝑥 ∈ 𝑈) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅)
5	3, 4	syl6 35	. . . 4 ⊢ (𝑋 ∈ 𝑈 → ((𝑛 = 1o ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅))
6	5	imp 406	. . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅)
7		1onn 8678	. . . 4 ⊢ 1o ∈ ω
8	7	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑈 → 1o ∈ ω)
9		elex 3501	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V)
10		0ex 5307	. . . 4 ⊢ ∅ ∈ V
11	10	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑈 → ∅ ∈ V)
12	1, 6, 8, 9, 11	ovmpod 7585	. 2 ⊢ (𝑋 ∈ 𝑈 → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ∅)
13		df-ov 7434	. 2 ⊢ (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩)
14	12, 13	eqtr3di 2792	1 ⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  ifcif 4525  ⟨cop 4632  ∪ cuni 4907   × cxp 5683  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  ωcom 7887  1^st c1st 8012  1_oc1o 8499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1o 8506

This theorem is referenced by:  finxp1o  37393