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Theorem finxpreclem5 33203
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem5.1 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
Assertion
Ref Expression
finxpreclem5 ((𝑛 ∈ ω ∧ 1𝑜𝑛) → (¬ 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
StepHypRef Expression
1 df-ov 6638 . . 3 (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2 vex 3198 . . . . . 6 𝑥 ∈ V
3 0ex 4781 . . . . . . 7 ∅ ∈ V
4 opex 4923 . . . . . . . 8 𝑛, (1st𝑥)⟩ ∈ V
5 opex 4923 . . . . . . . 8 𝑛, 𝑥⟩ ∈ V
64, 5ifex 4147 . . . . . . 7 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
73, 6ifex 4147 . . . . . 6 if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V
8 finxpreclem5.1 . . . . . . 7 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
98ovmpt4g 6768 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑥 ∈ V ∧ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) ∈ V) → (𝑛𝐹𝑥) = if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
102, 7, 9mp3an23 1414 . . . . 5 (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
1110ad2antrr 761 . . . 4 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
12 1on 7552 . . . . . . . . . . 11 1𝑜 ∈ On
1312onirri 5822 . . . . . . . . . 10 ¬ 1𝑜 ∈ 1𝑜
14 eleq2 2688 . . . . . . . . . 10 (𝑛 = 1𝑜 → (1𝑜𝑛 ↔ 1𝑜 ∈ 1𝑜))
1513, 14mtbiri 317 . . . . . . . . 9 (𝑛 = 1𝑜 → ¬ 1𝑜𝑛)
1615con2i 134 . . . . . . . 8 (1𝑜𝑛 → ¬ 𝑛 = 1𝑜)
1716intnanrd 962 . . . . . . 7 (1𝑜𝑛 → ¬ (𝑛 = 1𝑜𝑥𝑈))
1817iffalsed 4088 . . . . . 6 (1𝑜𝑛 → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
1918adantl 482 . . . . 5 ((𝑛 ∈ ω ∧ 1𝑜𝑛) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
20 iffalse 4086 . . . . 5 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2119, 20sylan9eq 2674 . . . 4 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨𝑛, 𝑥⟩)
2211, 21eqtrd 2654 . . 3 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
231, 22syl5eqr 2668 . 2 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2423ex 450 1 ((𝑛 ∈ ω ∧ 1𝑜𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  ifcif 4077  cop 4174   cuni 4427   × cxp 5102  cfv 5876  (class class class)co 6635  cmpt2 6637  ωcom 7050  1st c1st 7151  1𝑜c1o 7538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1o 7545
This theorem is referenced by:  finxpreclem6  33204
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