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Theorem finxpreclem3 33201
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 6638 . 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 2624 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 = 1𝑜𝑁 = 1𝑜))
5 eleq1 2687 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
64, 5bi2anan9 916 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → ((𝑛 = 1𝑜𝑥𝑈) ↔ (𝑁 = 1𝑜𝑋𝑈)))
7 eleq1 2687 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
87adantl 482 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
9 unieq 4435 . . . . . . . . 9 (𝑛 = 𝑁 𝑛 = 𝑁)
109adantr 481 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
11 fveq2 6178 . . . . . . . . 9 (𝑥 = 𝑋 → (1st𝑥) = (1st𝑋))
1211adantl 482 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → (1st𝑥) = (1st𝑋))
1310, 12opeq12d 4401 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨ 𝑛, (1st𝑥)⟩ = ⟨ 𝑁, (1st𝑋)⟩)
14 opeq12 4395 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
158, 13, 14ifbieq12d 4104 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
166, 15ifbieq2d 4102 . . . . 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 5790 . . . . . . . . . . . . 13 1𝑜 ⊆ suc 1𝑜
18 df-2o 7546 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
1917, 18sseqtr4i 3630 . . . . . . . . . . . 12 1𝑜 ⊆ 2𝑜
20 1on 7552 . . . . . . . . . . . . . 14 1𝑜 ∈ On
2118, 20sucneqoni 33185 . . . . . . . . . . . . 13 2𝑜 ≠ 1𝑜
2221necomi 2845 . . . . . . . . . . . 12 1𝑜 ≠ 2𝑜
23 df-pss 3583 . . . . . . . . . . . 12 (1𝑜 ⊊ 2𝑜 ↔ (1𝑜 ⊆ 2𝑜 ∧ 1𝑜 ≠ 2𝑜))
2419, 22, 23mpbir2an 954 . . . . . . . . . . 11 1𝑜 ⊊ 2𝑜
25 ssnpss 3702 . . . . . . . . . . 11 (2𝑜 ⊆ 1𝑜 → ¬ 1𝑜 ⊊ 2𝑜)
2624, 25mt2 191 . . . . . . . . . 10 ¬ 2𝑜 ⊆ 1𝑜
27 sseq2 3619 . . . . . . . . . 10 (𝑁 = 1𝑜 → (2𝑜𝑁 ↔ 2𝑜 ⊆ 1𝑜))
2826, 27mtbiri 317 . . . . . . . . 9 (𝑁 = 1𝑜 → ¬ 2𝑜𝑁)
2928con2i 134 . . . . . . . 8 (2𝑜𝑁 → ¬ 𝑁 = 1𝑜)
3029intnanrd 962 . . . . . . 7 (2𝑜𝑁 → ¬ (𝑁 = 1𝑜𝑋𝑈))
3130iffalsed 4088 . . . . . 6 (2𝑜𝑁 → if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
32 iftrue 4083 . . . . . 6 (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨ 𝑁, (1st𝑋)⟩)
3331, 32sylan9eq 2674 . . . . 5 ((2𝑜𝑁𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3416, 33sylan9eqr 2676 . . . 4 (((2𝑜𝑁𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3534adantlll 753 . . 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 789 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
37 elex 3207 . . . 4 (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
3837adantl 482 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
39 opex 4923 . . . 4 𝑁, (1st𝑋)⟩ ∈ V
4039a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ ∈ V)
413, 35, 36, 38, 40ovmpt2d 6773 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨ 𝑁, (1st𝑋)⟩)
421, 41syl5reqr 2669 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 384   = wceq 1481  wcel 1988  wne 2791  Vcvv 3195  wss 3567  wpss 3568  c0 3907  ifcif 4077  cop 4174   cuni 4427   × cxp 5102  suc csuc 5713  cfv 5876  (class class class)co 6635  cmpt2 6637  ωcom 7050  1st c1st 7151  1𝑜c1o 7538  2𝑜c2o 7539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1o 7545  df-2o 7546
This theorem is referenced by:  finxpreclem4  33202
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