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Theorem finxpreclem3 33560
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem3.1 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
Assertion
Ref Expression
finxpreclem3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))
Distinct variable groups:   𝑛,𝑁,𝑥   𝑈,𝑛,𝑥   𝑛,𝑋,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑛)

Proof of Theorem finxpreclem3
StepHypRef Expression
1 df-ov 6795 . 2 (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
2 finxpreclem3.1 . . . 4 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
32a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))))
4 eqeq1 2774 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 = 1𝑜𝑁 = 1𝑜))
5 eleq1 2837 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
64, 5bi2anan9 612 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → ((𝑛 = 1𝑜𝑥𝑈) ↔ (𝑁 = 1𝑜𝑋𝑈)))
7 eleq1 2837 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
87adantl 467 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
9 unieq 4580 . . . . . . . . 9 (𝑛 = 𝑁 𝑛 = 𝑁)
109adantr 466 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
11 fveq2 6332 . . . . . . . . 9 (𝑥 = 𝑋 → (1st𝑥) = (1st𝑋))
1211adantl 467 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → (1st𝑥) = (1st𝑋))
1310, 12opeq12d 4545 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨ 𝑛, (1st𝑥)⟩ = ⟨ 𝑁, (1st𝑋)⟩)
14 opeq12 4539 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
158, 13, 14ifbieq12d 4250 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
166, 15ifbieq2d 4248 . . . . 5 ((𝑛 = 𝑁𝑥 = 𝑋) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)))
17 sssucid 5945 . . . . . . . . . . . . 13 1𝑜 ⊆ suc 1𝑜
18 df-2o 7713 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
1917, 18sseqtr4i 3785 . . . . . . . . . . . 12 1𝑜 ⊆ 2𝑜
20 1on 7719 . . . . . . . . . . . . . 14 1𝑜 ∈ On
2118, 20sucneqoni 33544 . . . . . . . . . . . . 13 2𝑜 ≠ 1𝑜
2221necomi 2996 . . . . . . . . . . . 12 1𝑜 ≠ 2𝑜
23 df-pss 3737 . . . . . . . . . . . 12 (1𝑜 ⊊ 2𝑜 ↔ (1𝑜 ⊆ 2𝑜 ∧ 1𝑜 ≠ 2𝑜))
2419, 22, 23mpbir2an 682 . . . . . . . . . . 11 1𝑜 ⊊ 2𝑜
25 ssnpss 3858 . . . . . . . . . . 11 (2𝑜 ⊆ 1𝑜 → ¬ 1𝑜 ⊊ 2𝑜)
2624, 25mt2 191 . . . . . . . . . 10 ¬ 2𝑜 ⊆ 1𝑜
27 sseq2 3774 . . . . . . . . . 10 (𝑁 = 1𝑜 → (2𝑜𝑁 ↔ 2𝑜 ⊆ 1𝑜))
2826, 27mtbiri 316 . . . . . . . . 9 (𝑁 = 1𝑜 → ¬ 2𝑜𝑁)
2928con2i 136 . . . . . . . 8 (2𝑜𝑁 → ¬ 𝑁 = 1𝑜)
3029intnanrd 999 . . . . . . 7 (2𝑜𝑁 → ¬ (𝑁 = 1𝑜𝑋𝑈))
3130iffalsed 4234 . . . . . 6 (2𝑜𝑁 → if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
32 iftrue 4229 . . . . . 6 (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨ 𝑁, (1st𝑋)⟩)
3331, 32sylan9eq 2824 . . . . 5 ((2𝑜𝑁𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3416, 33sylan9eqr 2826 . . . 4 (((2𝑜𝑁𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3534adantlll 689 . . 3 ((((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
36 simpll 742 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
37 elex 3361 . . . 4 (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
3837adantl 467 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
39 opex 5060 . . . 4 𝑁, (1st𝑋)⟩ ∈ V
4039a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ ∈ V)
413, 35, 36, 38, 40ovmpt2d 6934 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨ 𝑁, (1st𝑋)⟩)
421, 41syl5reqr 2819 1 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942  Vcvv 3349  wss 3721  wpss 3722  c0 4061  ifcif 4223  cop 4320   cuni 4572   × cxp 5247  suc csuc 5868  cfv 6031  (class class class)co 6792  cmpt2 6794  ωcom 7211  1st c1st 7312  1𝑜c1o 7705  2𝑜c2o 7706
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  df-2o 7713
This theorem is referenced by:  finxpreclem4  33561
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