Theorem gdlop2fv

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

	Ref	Expression
Hypotheses	gdlop2fv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlop2fv.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlop2fv	⊢ ( 𝜑 → ( ℱ2 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ E ) )

Proof

Step	Hyp	Ref	Expression
1		gdlop2fv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlop2fv.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
4	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
5		inex1g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∩ E ) ∈ V )
6	1 5	syl	⊢ ( 𝜑 → ( 𝐴 ∩ E ) ∈ V )
7		ineq1	⊢ ( 𝐴 = 𝑥 → ( 𝐴 ∩ E ) = ( 𝑥 ∩ E ) )
8	7	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( 𝐴 ∩ E ) = ( 𝑥 ∩ E ) )
9		eqidd	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( 𝐴 ∩ E ) = ( 𝐴 ∩ E ) )
10		df-gdlop2	⊢ ℱ2 = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∩ E ) )
11	3 4 6 8 9 10	mpofvop2d	⊢ ( 𝜑 → ( ℱ2 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ E ) )