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Theorem finxpreclem3 36578
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 2735 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 = 1o𝑁 = 1o))
4 eleq1 2820 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
53, 4bi2anan9 636 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → ((𝑛 = 1o𝑥𝑈) ↔ (𝑁 = 1o𝑋𝑈)))
6 eleq1 2820 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
76adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
8 unieq 4919 . . . . . . . . 9 (𝑛 = 𝑁 𝑛 = 𝑁)
98adantr 480 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
10 fveq2 6891 . . . . . . . . 9 (𝑥 = 𝑋 → (1st𝑥) = (1st𝑋))
1110adantl 481 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → (1st𝑥) = (1st𝑋))
129, 11opeq12d 4881 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨ 𝑛, (1st𝑥)⟩ = ⟨ 𝑁, (1st𝑋)⟩)
13 opeq12 4875 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
147, 12, 13ifbieq12d 4556 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
155, 14ifbieq2d 4554 . . . . 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 6444 . . . . . . . . . . . . 13 1o ⊆ suc 1o
17 df-2o 8471 . . . . . . . . . . . . 13 2o = suc 1o
1816, 17sseqtrri 4019 . . . . . . . . . . . 12 1o ⊆ 2o
19 1on 8482 . . . . . . . . . . . . . 14 1o ∈ On
2017, 19sucneqoni 36551 . . . . . . . . . . . . 13 2o ≠ 1o
2120necomi 2994 . . . . . . . . . . . 12 1o ≠ 2o
22 df-pss 3967 . . . . . . . . . . . 12 (1o ⊊ 2o ↔ (1o ⊆ 2o ∧ 1o ≠ 2o))
2318, 21, 22mpbir2an 708 . . . . . . . . . . 11 1o ⊊ 2o
24 ssnpss 4103 . . . . . . . . . . 11 (2o ⊆ 1o → ¬ 1o ⊊ 2o)
2523, 24mt2 199 . . . . . . . . . 10 ¬ 2o ⊆ 1o
26 sseq2 4008 . . . . . . . . . 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 4539 . . . . . 6 (2o𝑁 → if((𝑁 = 1o𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
31 iftrue 4534 . . . . . 6 (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨ 𝑁, (1st𝑋)⟩)
3230, 31sylan9eq 2791 . . . . 5 ((2o𝑁𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1o𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3315, 32sylan9eqr 2793 . . . 4 (((2o𝑁𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3433adantlll 715 . . 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 764 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
36 elex 3492 . . . 4 (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
3736adantl 481 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
38 opex 5464 . . . 4 𝑁, (1st𝑋)⟩ ∈ V
3938a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ ∈ V)
402, 34, 35, 37, 39ovmpod 7563 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨ 𝑁, (1st𝑋)⟩)
41 df-ov 7415 . 2 (𝑁𝐹𝑋) = (𝐹‘⟨𝑁, 𝑋⟩)
4240, 41eqtr3di 2786 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 395   = wceq 1540  wcel 2105  wne 2939  Vcvv 3473  wss 3948  wpss 3949  c0 4322  ifcif 4528  cop 4634   cuni 4908   × cxp 5674  suc csuc 6366  cfv 6543  (class class class)co 7412  cmpo 7414  ωcom 7859  1st c1st 7977  1oc1o 8463  2oc2o 8464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1o 8470  df-2o 8471
This theorem is referenced by:  finxpreclem4  36579
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