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Theorem finxpreclem3 34543
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 7154 . 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 2829 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 = 1o𝑁 = 1o))
5 eleq1 2904 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
64, 5bi2anan9 635 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → ((𝑛 = 1o𝑥𝑈) ↔ (𝑁 = 1o𝑋𝑈)))
7 eleq1 2904 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
87adantl 482 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
9 unieq 4844 . . . . . . . . 9 (𝑛 = 𝑁 𝑛 = 𝑁)
109adantr 481 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
11 fveq2 6666 . . . . . . . . 9 (𝑥 = 𝑋 → (1st𝑥) = (1st𝑋))
1211adantl 482 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → (1st𝑥) = (1st𝑋))
1310, 12opeq12d 4809 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨ 𝑛, (1st𝑥)⟩ = ⟨ 𝑁, (1st𝑋)⟩)
14 opeq12 4803 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
158, 13, 14ifbieq12d 4496 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
166, 15ifbieq2d 4494 . . . . 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 6265 . . . . . . . . . . . . 13 1o ⊆ suc 1o
18 df-2o 8097 . . . . . . . . . . . . 13 2o = suc 1o
1917, 18sseqtrri 4007 . . . . . . . . . . . 12 1o ⊆ 2o
20 1on 8103 . . . . . . . . . . . . . 14 1o ∈ On
2118, 20sucneqoni 34516 . . . . . . . . . . . . 13 2o ≠ 1o
2221necomi 3074 . . . . . . . . . . . 12 1o ≠ 2o
23 df-pss 3957 . . . . . . . . . . . 12 (1o ⊊ 2o ↔ (1o ⊆ 2o ∧ 1o ≠ 2o))
2419, 22, 23mpbir2an 707 . . . . . . . . . . 11 1o ⊊ 2o
25 ssnpss 4083 . . . . . . . . . . 11 (2o ⊆ 1o → ¬ 1o ⊊ 2o)
2624, 25mt2 201 . . . . . . . . . 10 ¬ 2o ⊆ 1o
27 sseq2 3996 . . . . . . . . . 10 (𝑁 = 1o → (2o𝑁 ↔ 2o ⊆ 1o))
2826, 27mtbiri 328 . . . . . . . . 9 (𝑁 = 1o → ¬ 2o𝑁)
2928con2i 141 . . . . . . . 8 (2o𝑁 → ¬ 𝑁 = 1o)
3029intnanrd 490 . . . . . . 7 (2o𝑁 → ¬ (𝑁 = 1o𝑋𝑈))
3130iffalsed 4480 . . . . . 6 (2o𝑁 → if((𝑁 = 1o𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
32 iftrue 4475 . . . . . 6 (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨ 𝑁, (1st𝑋)⟩)
3331, 32sylan9eq 2880 . . . . 5 ((2o𝑁𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1o𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3416, 33sylan9eqr 2882 . . . 4 (((2o𝑁𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3534adantlll 714 . . 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 763 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
37 elex 3517 . . . 4 (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
3837adantl 482 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
39 opex 5352 . . . 4 𝑁, (1st𝑋)⟩ ∈ V
4039a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ ∈ V)
413, 35, 36, 38, 40ovmpod 7295 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨ 𝑁, (1st𝑋)⟩)
421, 41syl5reqr 2875 1 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3020  Vcvv 3499  wss 3939  wpss 3940  c0 4294  ifcif 4469  cop 4569   cuni 4836   × cxp 5551  suc csuc 6190  cfv 6351  (class class class)co 7151  cmpo 7153  ωcom 7571  1st c1st 7681  1oc1o 8089  2oc2o 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-ord 6191  df-on 6192  df-suc 6194  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1o 8096  df-2o 8097
This theorem is referenced by:  finxpreclem4  34544
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