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

Theorem gonafv 35356
Description: The "Godel-set for the Sheffer stroke NAND" for two formulas 𝐴 and 𝐵. (Contributed by AV, 16-Oct-2023.)
Assertion
Ref Expression
gonafv ((𝐴𝑉𝐵𝑊) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)

Proof of Theorem gonafv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7435 . 2 (𝐴𝑔𝐵) = (⊼𝑔‘⟨𝐴, 𝐵⟩)
2 opelvvg 5725 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3 opeq2 4873 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → ⟨1o, 𝑥⟩ = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
4 df-gona 35347 . . . 4 𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
5 opex 5468 . . . 4 ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V
63, 4, 5fvmpt 7015 . . 3 (⟨𝐴, 𝐵⟩ ∈ (V × V) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
72, 6syl 17 . 2 ((𝐴𝑉𝐵𝑊) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
81, 7eqtrid 2788 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cop 4631   × cxp 5682  cfv 6560  (class class class)co 7432  1oc1o 8500  𝑔cgna 35340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-gona 35347
This theorem is referenced by:  gonanegoal  35358  fmlaomn0  35396  gonan0  35398  gonarlem  35400  gonar  35401  fmla0disjsuc  35404  fmlasucdisj  35405  satffunlem  35407  satffunlem1lem1  35408  satffunlem2lem1  35410
  Copyright terms: Public domain W3C validator