Theorem gdlop5fv

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

	Ref	Expression
Hypotheses	gdlop5fv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlop5fv.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlop5fv	⊢ ( 𝜑 → ( ℱ5 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ dom 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gdlop5fv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlop5fv.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
4	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
5		inex1g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∩ dom 𝐵 ) ∈ V )
6	1 5	syl	⊢ ( 𝜑 → ( 𝐴 ∩ dom 𝐵 ) ∈ V )
7		ineq1	⊢ ( 𝐴 = 𝑥 → ( 𝐴 ∩ dom 𝑦 ) = ( 𝑥 ∩ dom 𝑦 ) )
8	7	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( 𝐴 ∩ dom 𝑦 ) = ( 𝑥 ∩ dom 𝑦 ) )
9		dmeq	⊢ ( 𝐵 = 𝑦 → dom 𝐵 = dom 𝑦 )
10	9	ineq2d	⊢ ( 𝐵 = 𝑦 → ( 𝐴 ∩ dom 𝐵 ) = ( 𝐴 ∩ dom 𝑦 ) )
11	10	adantl	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( 𝐴 ∩ dom 𝐵 ) = ( 𝐴 ∩ dom 𝑦 ) )
12		df-gdlop5	⊢ ℱ5 = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∩ dom 𝑦 ) )
13	3 4 6 8 11 12	mpofvop2d	⊢ ( 𝜑 → ( ℱ5 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∩ dom 𝐵 ) )