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 33048
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 7235 . 2 (𝐴𝑔𝐵) = (⊼𝑔‘⟨𝐴, 𝐵⟩)
2 opelvvg 5606 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3 opeq2 4800 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → ⟨1o, 𝑥⟩ = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
4 df-gona 33039 . . . 4 𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
5 opex 5363 . . . 4 ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V
63, 4, 5fvmpt 6837 . . 3 (⟨𝐴, 𝐵⟩ ∈ (V × V) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
72, 6syl 17 . 2 ((𝐴𝑉𝐵𝑊) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
81, 7syl5eq 2791 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  Vcvv 3421  cop 4562   × cxp 5564  cfv 6398  (class class class)co 7232  1oc1o 8216  𝑔cgna 33032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-iota 6356  df-fun 6400  df-fv 6406  df-ov 7235  df-gona 33039
This theorem is referenced by:  gonanegoal  33050  fmlaomn0  33088  gonan0  33090  gonarlem  33092  gonar  33093  fmla0disjsuc  33096  fmlasucdisj  33097  satffunlem  33099  satffunlem1lem1  33100  satffunlem2lem1  33102
  Copyright terms: Public domain W3C validator