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Theorem finxpreclem3 37376
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 finxpreclem3.1 . . . 4 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
21a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))))
3 eqeq1 2739 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 = 1o𝑁 = 1o))
4 eleq1 2827 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
53, 4bi2anan9 638 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → ((𝑛 = 1o𝑥𝑈) ↔ (𝑁 = 1o𝑋𝑈)))
6 eleq1 2827 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
76adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8 unieq 4923 . . . . . . . . 9 (𝑛 = 𝑁 𝑛 = 𝑁)
98adantr 480 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
10 fveq2 6907 . . . . . . . . 9 (𝑥 = 𝑋 → (1st𝑥) = (1st𝑋))
1110adantl 481 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → (1st𝑥) = (1st𝑋))
129, 11opeq12d 4886 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨ 𝑛, (1st𝑥)⟩ = ⟨ 𝑁, (1st𝑋)⟩)
13 opeq12 4880 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
147, 12, 13ifbieq12d 4559 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
155, 14ifbieq2d 4557 . . . . 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⟩)))
16 sssucid 6466 . . . . . . . . . . . . 13 1o ⊆ suc 1o
17 df-2o 8506 . . . . . . . . . . . . 13 2o = suc 1o
1816, 17sseqtrri 4033 . . . . . . . . . . . 12 1o ⊆ 2o
19 1on 8517 . . . . . . . . . . . . . 14 1o ∈ On
2017, 19sucneqoni 37349 . . . . . . . . . . . . 13 2o ≠ 1o
2120necomi 2993 . . . . . . . . . . . 12 1o ≠ 2o
22 df-pss 3983 . . . . . . . . . . . 12 (1o ⊊ 2o ↔ (1o ⊆ 2o ∧ 1o ≠ 2o))
2318, 21, 22mpbir2an 711 . . . . . . . . . . 11 1o ⊊ 2o
24 ssnpss 4116 . . . . . . . . . . 11 (2o ⊆ 1o → ¬ 1o ⊊ 2o)
2523, 24mt2 200 . . . . . . . . . 10 ¬ 2o ⊆ 1o
26 sseq2 4022 . . . . . . . . . 10 (𝑁 = 1o → (2o𝑁 ↔ 2o ⊆ 1o))
2725, 26mtbiri 327 . . . . . . . . 9 (𝑁 = 1o → ¬ 2o𝑁)
2827con2i 139 . . . . . . . 8 (2o𝑁 → ¬ 𝑁 = 1o)
2928intnanrd 489 . . . . . . 7 (2o𝑁 → ¬ (𝑁 = 1o𝑋𝑈))
3029iffalsed 4542 . . . . . 6 (2o𝑁 → if((𝑁 = 1o𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
31 iftrue 4537 . . . . . 6 (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨ 𝑁, (1st𝑋)⟩)
3230, 31sylan9eq 2795 . . . . 5 ((2o𝑁𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1o𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3315, 32sylan9eqr 2797 . . . 4 (((2o𝑁𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3433adantlll 718 . . 3 ((((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
35 simpll 767 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
36 elex 3499 . . . 4 (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
3736adantl 481 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
38 opex 5475 . . . 4 𝑁, (1st𝑋)⟩ ∈ V
3938a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ ∈ V)
402, 34, 35, 37, 39ovmpod 7585 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨ 𝑁, (1st𝑋)⟩)
41 df-ov 7434 . 2 (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
4240, 41eqtr3di 2790 1 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  wss 3963  wpss 3964  c0 4339  ifcif 4531  cop 4637   cuni 4912   × cxp 5687  suc csuc 6388  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  1st c1st 8011  1oc1o 8498  2oc2o 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506
This theorem is referenced by:  finxpreclem4  37377
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