Theorem gonafv 35310

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 7372	. 2 ⊢ (𝐴⊼𝑔𝐵) = (⊼𝑔‘⟨𝐴, 𝐵⟩)
2		opelvvg 5672	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3		opeq2 4834	. . . 4 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ⟨1o, 𝑥⟩ = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
4		df-gona 35301	. . . 4 ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
5		opex 5419	. . . 4 ⊢ ⟨1o, ⟨𝐴, 𝐵⟩⟩ ∈ V
6	3, 4, 5	fvmpt 6950	. . 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 2776	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   × cxp 5629  ‘cfv 6499  (class class class)co 7369  1_oc1o 8404  ⊼_𝑔cgna 35294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-gona 35301

This theorem is referenced by:  gonanegoal  35312  fmlaomn0  35350  gonan0  35352  gonarlem  35354  gonar  35355  fmla0disjsuc  35358  fmlasucdisj  35359  satffunlem  35361  satffunlem1lem1  35362  satffunlem2lem1  35364