Theorem finxpreclem1 37594

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 2737	. . 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 2831	. . . . . 6 ⊢ (𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → 𝑥 ∈ 𝑈))
3	2	anim2d 612	. . . . 5 ⊢ (𝑋 ∈ 𝑈 → ((𝑛 = 1o ∧ 𝑥 = 𝑋) → (𝑛 = 1o ∧ 𝑥 ∈ 𝑈)))
4		iftrue 4485	. . . . 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 8568	. . . 4 ⊢ 1o ∈ ω
8	7	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑈 → 1o ∈ ω)
9		elex 3461	. . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V)
10		0ex 5252	. . . 4 ⊢ ∅ ∈ V
11	10	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑈 → ∅ ∈ V)
12	1, 6, 8, 9, 11	ovmpod 7510	. 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 7361	. 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 2786	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 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  ifcif 4479  ⟨cop 4586  ∪ cuni 4863   × cxp 5622  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  ωcom 7808  1^st c1st 7931  1_oc1o 8390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1o 8397

This theorem is referenced by:  finxp1o  37597