Theorem gonafv 35318

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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)

Proof of Theorem gonafv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7451	. 2 ⊢ (𝐴⊼𝑔𝐵) = (⊼𝑔‘⟨𝐴, 𝐵⟩)
2		opelvvg 5741	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3		opeq2 4898	. . . 4 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ⟨1o, 𝑥⟩ = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
4		df-gona 35309	. . . 4 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
5		opex 5484	. . . 4 ⊢ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V
6	3, 4, 5	fvmpt 7029	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (V × V) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
7	2, 6	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⊼𝑔‘⟨𝐴, 𝐵⟩) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
8	1, 7	eqtrid 2792	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654   × cxp 5698  ‘cfv 6573  (class class class)co 7448  1_oc1o 8515  ⊼_𝑔cgna 35302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-gona 35309

This theorem is referenced by:  gonanegoal  35320  fmlaomn0  35358  gonan0  35360  gonarlem  35362  gonar  35363  fmla0disjsuc  35366  fmlasucdisj  35367  satffunlem  35369  satffunlem1lem1  35370  satffunlem2lem1  35372