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Theorem finxpreclem5 35563
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 7280 . . 3 (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2 vex 3435 . . . . . 6 𝑥 ∈ V
3 0ex 5233 . . . . . . 7 ∅ ∈ V
4 opex 5381 . . . . . . . 8 𝑛, (1st𝑥)⟩ ∈ V
5 opex 5381 . . . . . . . 8 𝑛, 𝑥⟩ ∈ V
64, 5ifex 4511 . . . . . . 7 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
73, 6ifex 4511 . . . . . 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 7420 . . . . . 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 1452 . . . . 5 (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
1110ad2antrr 723 . . . 4 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
12 1on 8307 . . . . . . . . . . 11 1o ∈ On
1312onirri 6375 . . . . . . . . . 10 ¬ 1o ∈ 1o
14 eleq2 2827 . . . . . . . . . 10 (𝑛 = 1o → (1o𝑛 ↔ 1o ∈ 1o))
1513, 14mtbiri 327 . . . . . . . . 9 (𝑛 = 1o → ¬ 1o𝑛)
1615con2i 139 . . . . . . . 8 (1o𝑛 → ¬ 𝑛 = 1o)
1716intnanrd 490 . . . . . . 7 (1o𝑛 → ¬ (𝑛 = 1o𝑥𝑈))
1817iffalsed 4472 . . . . . 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 4470 . . . . 5 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2119, 20sylan9eq 2798 . . . 4 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨𝑛, 𝑥⟩)
2211, 21eqtrd 2778 . . 3 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
231, 22eqtr3id 2792 . 2 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2423ex 413 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 396   = wceq 1539  wcel 2106  Vcvv 3431  c0 4258  ifcif 4461  cop 4569   cuni 4841   × cxp 5589  cfv 6435  (class class class)co 7277  cmpo 7279  ωcom 7712  1st c1st 7829  1oc1o 8288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-tr 5194  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-ord 6271  df-on 6272  df-suc 6274  df-iota 6393  df-fun 6437  df-fv 6443  df-ov 7280  df-oprab 7281  df-mpo 7282  df-1o 8295
This theorem is referenced by:  finxpreclem6  35564
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