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Theorem finxpreclem5 36579
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 7414 . . 3 (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2 vex 3476 . . . . . 6 𝑥 ∈ V
3 0ex 5306 . . . . . . 7 ∅ ∈ V
4 opex 5463 . . . . . . . 8 𝑛, (1st𝑥)⟩ ∈ V
5 opex 5463 . . . . . . . 8 𝑛, 𝑥⟩ ∈ V
64, 5ifex 4577 . . . . . . 7 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
73, 6ifex 4577 . . . . . 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 7557 . . . . . 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 1451 . . . . 5 (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
1110ad2antrr 722 . . . 4 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
12 1on 8480 . . . . . . . . . . 11 1o ∈ On
1312onirri 6476 . . . . . . . . . 10 ¬ 1o ∈ 1o
14 eleq2 2820 . . . . . . . . . 10 (𝑛 = 1o → (1o𝑛 ↔ 1o ∈ 1o))
1513, 14mtbiri 326 . . . . . . . . 9 (𝑛 = 1o → ¬ 1o𝑛)
1615con2i 139 . . . . . . . 8 (1o𝑛 → ¬ 𝑛 = 1o)
1716intnanrd 488 . . . . . . 7 (1o𝑛 → ¬ (𝑛 = 1o𝑥𝑈))
1817iffalsed 4538 . . . . . 6 (1o𝑛 → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
1918adantl 480 . . . . 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 4536 . . . . 5 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2119, 20sylan9eq 2790 . . . 4 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨𝑛, 𝑥⟩)
2211, 21eqtrd 2770 . . 3 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
231, 22eqtr3id 2784 . 2 (((𝑛 ∈ ω ∧ 1o𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2423ex 411 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 394   = wceq 1539  wcel 2104  Vcvv 3472  c0 4321  ifcif 4527  cop 4633   cuni 4907   × cxp 5673  cfv 6542  (class class class)co 7411  cmpo 7413  ωcom 7857  1st c1st 7975  1oc1o 8461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1o 8468
This theorem is referenced by:  finxpreclem6  36580
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