Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  finxpreclem3 Structured version   Visualization version   GIF version

Theorem finxpreclem3 33568
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 6797 . 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 2775 . . . . . . 7 (𝑛 = 𝑁 → (𝑛 = 1𝑜𝑁 = 1𝑜))
5 eleq1 2838 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑈𝑋𝑈))
64, 5bi2anan9 614 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → ((𝑛 = 1𝑜𝑥𝑈) ↔ (𝑁 = 1𝑜𝑋𝑈)))
7 eleq1 2838 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
87adantl 467 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → (𝑥 ∈ (V × 𝑈) ↔ 𝑋 ∈ (V × 𝑈)))
9 unieq 4583 . . . . . . . . 9 (𝑛 = 𝑁 𝑛 = 𝑁)
109adantr 466 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → 𝑛 = 𝑁)
11 fveq2 6333 . . . . . . . . 9 (𝑥 = 𝑋 → (1st𝑥) = (1st𝑋))
1211adantl 467 . . . . . . . 8 ((𝑛 = 𝑁𝑥 = 𝑋) → (1st𝑥) = (1st𝑋))
1310, 12opeq12d 4548 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨ 𝑛, (1st𝑥)⟩ = ⟨ 𝑁, (1st𝑋)⟩)
14 opeq12 4542 . . . . . . 7 ((𝑛 = 𝑁𝑥 = 𝑋) → ⟨𝑛, 𝑥⟩ = ⟨𝑁, 𝑋⟩)
158, 13, 14ifbieq12d 4253 . . . . . 6 ((𝑛 = 𝑁𝑥 = 𝑋) → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
166, 15ifbieq2d 4251 . . . . 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 5946 . . . . . . . . . . . . 13 1𝑜 ⊆ suc 1𝑜
18 df-2o 7715 . . . . . . . . . . . . 13 2𝑜 = suc 1𝑜
1917, 18sseqtr4i 3788 . . . . . . . . . . . 12 1𝑜 ⊆ 2𝑜
20 1on 7721 . . . . . . . . . . . . . 14 1𝑜 ∈ On
2118, 20sucneqoni 33552 . . . . . . . . . . . . 13 2𝑜 ≠ 1𝑜
2221necomi 2997 . . . . . . . . . . . 12 1𝑜 ≠ 2𝑜
23 df-pss 3740 . . . . . . . . . . . 12 (1𝑜 ⊊ 2𝑜 ↔ (1𝑜 ⊆ 2𝑜 ∧ 1𝑜 ≠ 2𝑜))
2419, 22, 23mpbir2an 684 . . . . . . . . . . 11 1𝑜 ⊊ 2𝑜
25 ssnpss 3861 . . . . . . . . . . 11 (2𝑜 ⊆ 1𝑜 → ¬ 1𝑜 ⊊ 2𝑜)
2624, 25mt2 191 . . . . . . . . . 10 ¬ 2𝑜 ⊆ 1𝑜
27 sseq2 3777 . . . . . . . . . 10 (𝑁 = 1𝑜 → (2𝑜𝑁 ↔ 2𝑜 ⊆ 1𝑜))
2826, 27mtbiri 316 . . . . . . . . 9 (𝑁 = 1𝑜 → ¬ 2𝑜𝑁)
2928con2i 136 . . . . . . . 8 (2𝑜𝑁 → ¬ 𝑁 = 1𝑜)
3029intnanrd 999 . . . . . . 7 (2𝑜𝑁 → ¬ (𝑁 = 1𝑜𝑋𝑈))
3130iffalsed 4237 . . . . . 6 (2𝑜𝑁 → if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩))
32 iftrue 4232 . . . . . 6 (𝑋 ∈ (V × 𝑈) → if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩) = ⟨ 𝑁, (1st𝑋)⟩)
3331, 32sylan9eq 2825 . . . . 5 ((2𝑜𝑁𝑋 ∈ (V × 𝑈)) → if((𝑁 = 1𝑜𝑋𝑈), ∅, if(𝑋 ∈ (V × 𝑈), ⟨ 𝑁, (1st𝑋)⟩, ⟨𝑁, 𝑋⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3416, 33sylan9eqr 2827 . . . 4 (((2𝑜𝑁𝑋 ∈ (V × 𝑈)) ∧ (𝑛 = 𝑁𝑥 = 𝑋)) → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = ⟨ 𝑁, (1st𝑋)⟩)
3534adantlll 691 . . 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 744 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑁 ∈ ω)
37 elex 3364 . . . 4 (𝑋 ∈ (V × 𝑈) → 𝑋 ∈ V)
3837adantl 467 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → 𝑋 ∈ V)
39 opex 5061 . . . 4 𝑁, (1st𝑋)⟩ ∈ V
4039a1i 11 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑋)⟩ ∈ V)
413, 35, 36, 38, 40ovmpt2d 6936 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → (𝑁𝐹𝑋) = ⟨ 𝑁, (1st𝑋)⟩)
421, 41syl5reqr 2820 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 382   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  wss 3724  wpss 3725  c0 4064  ifcif 4226  cop 4323   cuni 4575   × cxp 5248  suc csuc 5869  cfv 6032  (class class class)co 6794  cmpt2 6796  ωcom 7213  1st c1st 7314  1𝑜c1o 7707  2𝑜c2o 7708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-ord 5870  df-on 5871  df-suc 5873  df-iota 5995  df-fun 6034  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1o 7714  df-2o 7715
This theorem is referenced by:  finxpreclem4  33569
  Copyright terms: Public domain W3C validator