Theorem gdlop6fv

Description: The value of the sixth Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026)

	Ref	Expression
Hypotheses	gdlop6fv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlop6fv.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlop6fv	⊢ ( 𝜑 → ( ℱ6 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ ◡ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gdlop6fv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlop6fv.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
4	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
5		inex1g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∩ ◡ 𝐵 ) ∈ V )
6	1 5	syl	⊢ ( 𝜑 → ( 𝐴 ∩ ◡ 𝐵 ) ∈ V )
7		ineq1	⊢ ( 𝐴 = 𝑥 → ( 𝐴 ∩ ◡ 𝑦 ) = ( 𝑥 ∩ ◡ 𝑦 ) )
8	7	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( 𝐴 ∩ ◡ 𝑦 ) = ( 𝑥 ∩ ◡ 𝑦 ) )
9		cnveq	⊢ ( 𝐵 = 𝑦 → ◡ 𝐵 = ◡ 𝑦 )
10	9	ineq2d	⊢ ( 𝐵 = 𝑦 → ( 𝐴 ∩ ◡ 𝐵 ) = ( 𝐴 ∩ ◡ 𝑦 ) )
11	10	adantl	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( 𝐴 ∩ ◡ 𝐵 ) = ( 𝐴 ∩ ◡ 𝑦 ) )
12		df-gdlop6	⊢ ℱ6 = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∩ ◡ 𝑦 ) )
13	3 4 6 8 11 12	mpofvop2d	⊢ ( 𝜑 → ( ℱ6 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ ◡ 𝐵 ) )