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Theorem finxpreclem1 33533
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)
Assertion
Ref Expression
finxpreclem1 (𝑋𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩))
Distinct variable groups:   𝑈,𝑛,𝑥   𝑛,𝑋,𝑥

Proof of Theorem finxpreclem1
StepHypRef Expression
1 df-ov 6812 . 2 (1𝑜(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩)
2 eqidd 2757 . . 3 (𝑋𝑈 → (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))))
3 eleq1a 2830 . . . . . 6 (𝑋𝑈 → (𝑥 = 𝑋𝑥𝑈))
43anim2d 590 . . . . 5 (𝑋𝑈 → ((𝑛 = 1𝑜𝑥 = 𝑋) → (𝑛 = 1𝑜𝑥𝑈)))
5 iftrue 4232 . . . . 5 ((𝑛 = 1𝑜𝑥𝑈) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅)
64, 5syl6 35 . . . 4 (𝑋𝑈 → ((𝑛 = 1𝑜𝑥 = 𝑋) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅))
76imp 444 . . 3 ((𝑋𝑈 ∧ (𝑛 = 1𝑜𝑥 = 𝑋)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅)
8 1onn 7884 . . . 4 1𝑜 ∈ ω
98a1i 11 . . 3 (𝑋𝑈 → 1𝑜 ∈ ω)
10 elex 3348 . . 3 (𝑋𝑈𝑋 ∈ V)
11 0ex 4938 . . . 4 ∅ ∈ V
1211a1i 11 . . 3 (𝑋𝑈 → ∅ ∈ V)
132, 7, 9, 10, 12ovmpt2d 6949 . 2 (𝑋𝑈 → (1𝑜(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ∅)
141, 13syl5reqr 2805 1 (𝑋𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1𝑜, 𝑋⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  Vcvv 3336  c0 4054  ifcif 4226  cop 4323   cuni 4584   × cxp 5260  cfv 6045  (class class class)co 6809  cmpt2 6811  ωcom 7226  1st c1st 7327  1𝑜c1o 7718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1o 7725
This theorem is referenced by:  finxp1o  33536
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