Theorem gdlop3fv

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

	Ref	Expression
Hypotheses	gdlop3fv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gdlop3fv.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	gdlop3fv	⊢ ( 𝜑 → ( ℱ3 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∖ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gdlop3fv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gdlop3fv.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3	1	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
4	2	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
5	1	difexd	⊢ ( 𝜑 → ( 𝐴 ∖ 𝐵 ) ∈ V )
6		difeq1	⊢ ( 𝐴 = 𝑥 → ( 𝐴 ∖ 𝑦 ) = ( 𝑥 ∖ 𝑦 ) )
7	6	adantl	⊢ ( ( 𝜑 ∧ 𝐴 = 𝑥 ) → ( 𝐴 ∖ 𝑦 ) = ( 𝑥 ∖ 𝑦 ) )
8		difeq2	⊢ ( 𝐵 = 𝑦 → ( 𝐴 ∖ 𝐵 ) = ( 𝐴 ∖ 𝑦 ) )
9	8	adantl	⊢ ( ( 𝜑 ∧ 𝐵 = 𝑦 ) → ( 𝐴 ∖ 𝐵 ) = ( 𝐴 ∖ 𝑦 ) )
10		df-gdlop3	⊢ ℱ3 = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∖ 𝑦 ) )
11	3 4 5 7 9 10	mpofvop2d	⊢ ( 𝜑 → ( ℱ3 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ( 𝐴 ∖ 𝐵 ) )