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Theorem finxpreclem1 33580
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 6815 . 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 2775 . . 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 2848 . . . . . 6 (𝑋𝑈 → (𝑥 = 𝑋𝑥𝑈))
43anim2d 600 . . . . 5 (𝑋𝑈 → ((𝑛 = 1𝑜𝑥 = 𝑋) → (𝑛 = 1𝑜𝑥𝑈)))
5 iftrue 4241 . . . . 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 394 . . 3 ((𝑋𝑈 ∧ (𝑛 = 1𝑜𝑥 = 𝑋)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ∅)
8 1onn 7894 . . . 4 1𝑜 ∈ ω
98a1i 11 . . 3 (𝑋𝑈 → 1𝑜 ∈ ω)
10 elex 3369 . . 3 (𝑋𝑈𝑋 ∈ V)
11 0ex 4937 . . . 4 ∅ ∈ V
1211a1i 11 . . 3 (𝑋𝑈 → ∅ ∈ V)
132, 7, 9, 10, 12ovmpt2d 6956 . 2 (𝑋𝑈 → (1𝑜(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))𝑋) = ∅)
141, 13syl5reqr 2823 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 1634  wcel 2148  Vcvv 3355  c0 4073  ifcif 4235  cop 4332   cuni 4585   × cxp 5261  cfv 6042  (class class class)co 6812  cmpt2 6814  ωcom 7233  1st c1st 7334  1𝑜c1o 7727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fv 6050  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1o 7734
This theorem is referenced by:  finxp1o  33583
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