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Theorem finxpreclem3 32406
 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 6552 . 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 2614 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 = 1𝑜𝑁 = 1𝑜))
5 eleq1 2676 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
64, 5bi2anan9 913 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → ((𝑛 = 1𝑜𝑥𝑈) ↔ (𝑁 = 1𝑜𝑋𝑈)))
7 eleq1 2676 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
87adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
9 unieq 4380 . . . . . . . . 9 (𝑛 = 𝑁 𝑛 = 𝑁)
109adantr 480 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
11 fveq2 6103 . . . . . . . . 9 (𝑥 = 𝑋 → (1st𝑥) = (1st𝑋))
1211adantl 481 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → (1st𝑥) = (1st𝑋))
1310, 12opeq12d 4348 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨ 𝑛, (1st𝑥)⟩ = ⟨ 𝑁, (1st𝑋)⟩)
14 opeq12 4342 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
158, 13, 14ifbieq12d 4063 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
166, 15ifbieq2d 4061 . . . . 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 5719 . . . . . . . . . . . . 13 1𝑜 ⊆ suc 1𝑜
18 df-2o 7448 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
1917, 18sseqtr4i 3601 . . . . . . . . . . . 12 1𝑜 ⊆ 2𝑜
20 1on 7454 . . . . . . . . . . . . . 14 1𝑜 ∈ On
2118, 20sucneqoni 32390 . . . . . . . . . . . . 13 2𝑜 ≠ 1𝑜
2221necomi 2836 . . . . . . . . . . . 12 1𝑜 ≠ 2𝑜
23 df-pss 3556 . . . . . . . . . . . 12 (1𝑜 ⊊ 2𝑜 ↔ (1𝑜 ⊆ 2𝑜 ∧ 1𝑜 ≠ 2𝑜))
2419, 22, 23mpbir2an 957 . . . . . . . . . . 11 1𝑜 ⊊ 2𝑜
25 ssnpss 3672 . . . . . . . . . . 11 (2𝑜 ⊆ 1𝑜 → ¬ 1𝑜 ⊊ 2𝑜)
2624, 25mt2 190 . . . . . . . . . 10 ¬ 2𝑜 ⊆ 1𝑜
27 sseq2 3590 . . . . . . . . . 10 (𝑁 = 1𝑜 → (2𝑜𝑁 ↔ 2𝑜 ⊆ 1𝑜))
2826, 27mtbiri 316 . . . . . . . . 9 (𝑁 = 1𝑜 → ¬ 2𝑜𝑁)
2928con2i 133 . . . . . . . 8 (2𝑜𝑁 → ¬ 𝑁 = 1𝑜)
3029intnanrd 954 . . . . . . 7 (2𝑜𝑁 → ¬ (𝑁 = 1𝑜𝑋𝑈))
3130iffalsed 4047 . . . . . 6 (2𝑜𝑁 → if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
32 iftrue 4042 . . . . . 6 (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨ 𝑁, (1st𝑋)⟩)
3331, 32sylan9eq 2664 . . . . 5 ((2𝑜𝑁𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3416, 33sylan9eqr 2666 . . . 4 (((2𝑜𝑁𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3534adantlll 750 . . 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 786 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
37 elex 3185 . . . 4 (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
3837adantl 481 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
39 opex 4859 . . . 4 𝑁, (1st𝑋)⟩ ∈ V
4039a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ ∈ V)
413, 35, 36, 38, 40ovmpt2d 6686 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨ 𝑁, (1st𝑋)⟩)
421, 41syl5reqr 2659 1 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874  ifcif 4036  ⟨cop 4131  ∪ cuni 4372   × cxp 5036  suc csuc 5642  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  1st c1st 7057  1𝑜c1o 7440  2𝑜c2o 7441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-2o 7448 This theorem is referenced by:  finxpreclem4  32407
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