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Theorem finxpreclem5 33562
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 6795 . . 3 (𝑛𝐹𝑥) = (𝐹‘⟨𝑛, 𝑥⟩)
2 vex 3352 . . . . . 6 𝑥 ∈ V
3 0ex 4921 . . . . . . 7 ∅ ∈ V
4 opex 5060 . . . . . . . 8 𝑛, (1st𝑥)⟩ ∈ V
5 opex 5060 . . . . . . . 8 𝑛, 𝑥⟩ ∈ V
64, 5ifex 4293 . . . . . . 7 if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) ∈ V
73, 6ifex 4293 . . . . . 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 6929 . . . . . 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 1563 . . . . 5 (𝑛 ∈ ω → (𝑛𝐹𝑥) = if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
1110ad2antrr 697 . . . 4 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
12 1on 7719 . . . . . . . . . . 11 1𝑜 ∈ On
1312onirri 5977 . . . . . . . . . 10 ¬ 1𝑜 ∈ 1𝑜
14 eleq2 2838 . . . . . . . . . 10 (𝑛 = 1𝑜 → (1𝑜𝑛 ↔ 1𝑜 ∈ 1𝑜))
1513, 14mtbiri 316 . . . . . . . . 9 (𝑛 = 1𝑜 → ¬ 1𝑜𝑛)
1615con2i 136 . . . . . . . 8 (1𝑜𝑛 → ¬ 𝑛 = 1𝑜)
1716intnanrd 999 . . . . . . 7 (1𝑜𝑛 → ¬ (𝑛 = 1𝑜𝑥𝑈))
1817iffalsed 4234 . . . . . 6 (1𝑜𝑛 → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
1918adantl 467 . . . . 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 4232 . . . . 5 𝑥 ∈ (V × 𝑈) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2119, 20sylan9eq 2824 . . . 4 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨𝑛, 𝑥⟩)
2211, 21eqtrd 2804 . . 3 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝑛𝐹𝑥) = ⟨𝑛, 𝑥⟩)
231, 22syl5eqr 2818 . 2 (((𝑛 ∈ ω ∧ 1𝑜𝑛) ∧ ¬ 𝑥 ∈ (V × 𝑈)) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)
2423ex 397 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 382   = wceq 1630  wcel 2144  Vcvv 3349  c0 4061  ifcif 4223  cop 4320   cuni 4572   × cxp 5247  cfv 6031  (class class class)co 6792  cmpt2 6794  ωcom 7211  1st c1st 7312  1𝑜c1o 7705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1o 7712
This theorem is referenced by:  finxpreclem6  33563
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