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Theorem finxpreclem5 34663
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 7151 . . 3 (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2 vex 3496 . . . . . 6 𝑥 ∈ V
3 0ex 5202 . . . . . . 7 ∅ ∈ V
4 opex 5347 . . . . . . . 8 𝑛, (1st𝑥)⟩ ∈ V
5 opex 5347 . . . . . . . 8 𝑛, 𝑥⟩ ∈ V
64, 5ifex 4513 . . . . . . 7 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
73, 6ifex 4513 . . . . . 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 7289 . . . . . 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 1446 . . . . 5 (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
1110ad2antrr 724 . . . 4 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
12 1on 8101 . . . . . . . . . . 11 1o ∈ On
1312onirri 6290 . . . . . . . . . 10 ¬ 1o ∈ 1o
14 eleq2 2899 . . . . . . . . . 10 (𝑛 = 1o → (1o𝑛 ↔ 1o ∈ 1o))
1513, 14mtbiri 329 . . . . . . . . 9 (𝑛 = 1o → ¬ 1o𝑛)
1615con2i 141 . . . . . . . 8 (1o𝑛 → ¬ 𝑛 = 1o)
1716intnanrd 492 . . . . . . 7 (1o𝑛 → ¬ (𝑛 = 1o𝑥𝑈))
1817iffalsed 4476 . . . . . 6 (1o𝑛 → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
1918adantl 484 . . . . 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 4474 . . . . 5 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2119, 20sylan9eq 2874 . . . 4 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨𝑛, 𝑥⟩)
2211, 21eqtrd 2854 . . 3 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
231, 22syl5eqr 2868 . 2 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2423ex 415 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 398   = wceq 1530  wcel 2107  Vcvv 3493  c0 4289  ifcif 4465  cop 4565   cuni 4830   × cxp 5546  cfv 6348  (class class class)co 7148  cmpo 7150  ωcom 7572  1st c1st 7679  1oc1o 8087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1o 8094
This theorem is referenced by:  finxpreclem6  34664
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