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Theorem finxpreclem3 35893
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 2741 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 = 1o𝑁 = 1o))
4 eleq1 2826 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
53, 4bi2anan9 638 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → ((𝑛 = 1o𝑥𝑈) ↔ (𝑁 = 1o𝑋𝑈)))
6 eleq1 2826 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
76adantl 483 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8 unieq 4881 . . . . . . . . 9 (𝑛 = 𝑁 𝑛 = 𝑁)
98adantr 482 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
10 fveq2 6847 . . . . . . . . 9 (𝑥 = 𝑋 → (1st𝑥) = (1st𝑋))
1110adantl 483 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → (1st𝑥) = (1st𝑋))
129, 11opeq12d 4843 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨ 𝑛, (1st𝑥)⟩ = ⟨ 𝑁, (1st𝑋)⟩)
13 opeq12 4837 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
147, 12, 13ifbieq12d 4519 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
155, 14ifbieq2d 4517 . . . . 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 6402 . . . . . . . . . . . . 13 1o ⊆ suc 1o
17 df-2o 8418 . . . . . . . . . . . . 13 2o = suc 1o
1816, 17sseqtrri 3986 . . . . . . . . . . . 12 1o ⊆ 2o
19 1on 8429 . . . . . . . . . . . . . 14 1o ∈ On
2017, 19sucneqoni 35866 . . . . . . . . . . . . 13 2o ≠ 1o
2120necomi 2999 . . . . . . . . . . . 12 1o ≠ 2o
22 df-pss 3934 . . . . . . . . . . . 12 (1o ⊊ 2o ↔ (1o ⊆ 2o ∧ 1o ≠ 2o))
2318, 21, 22mpbir2an 710 . . . . . . . . . . 11 1o ⊊ 2o
24 ssnpss 4068 . . . . . . . . . . 11 (2o ⊆ 1o → ¬ 1o ⊊ 2o)
2523, 24mt2 199 . . . . . . . . . 10 ¬ 2o ⊆ 1o
26 sseq2 3975 . . . . . . . . . 10 (𝑁 = 1o → (2o𝑁 ↔ 2o ⊆ 1o))
2725, 26mtbiri 327 . . . . . . . . 9 (𝑁 = 1o → ¬ 2o𝑁)
2827con2i 139 . . . . . . . 8 (2o𝑁 → ¬ 𝑁 = 1o)
2928intnanrd 491 . . . . . . 7 (2o𝑁 → ¬ (𝑁 = 1o𝑋𝑈))
3029iffalsed 4502 . . . . . 6 (2o𝑁 → if((𝑁 = 1o𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
31 iftrue 4497 . . . . . 6 (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨ 𝑁, (1st𝑋)⟩)
3230, 31sylan9eq 2797 . . . . 5 ((2o𝑁𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1o𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3315, 32sylan9eqr 2799 . . . 4 (((2o𝑁𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3433adantlll 717 . . 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 766 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
36 elex 3466 . . . 4 (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
3736adantl 483 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
38 opex 5426 . . . 4 𝑁, (1st𝑋)⟩ ∈ V
3938a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ ∈ V)
402, 34, 35, 37, 39ovmpod 7512 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨ 𝑁, (1st𝑋)⟩)
41 df-ov 7365 . 2 (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
4240, 41eqtr3di 2792 1 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2944  Vcvv 3448  wss 3915  wpss 3916  c0 4287  ifcif 4491  cop 4597   cuni 4870   × cxp 5636  suc csuc 6324  cfv 6501  (class class class)co 7362  cmpo 7364  ωcom 7807  1st c1st 7924  1oc1o 8410  2oc2o 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1o 8417  df-2o 8418
This theorem is referenced by:  finxpreclem4  35894
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