Theorem gdlop3fv

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

	Ref	Expression
Hypotheses	gdlop3fv.1	|- ( ph -> A e. V )
	gdlop3fv.2	|- ( ph -> B e. W )
Assertion	gdlop3fv	|- ( ph -> ( ~F3 ` <. A , B >. ) = ( A \ B ) )

Proof

Step	Hyp	Ref	Expression
1		gdlop3fv.1	|- ( ph -> A e. V )
2		gdlop3fv.2	|- ( ph -> B e. W )
3	1	elexd	|- ( ph -> A e. _V )
4	2	elexd	|- ( ph -> B e. _V )
5	1	difexd	|- ( ph -> ( A \ B ) e. _V )
6		difeq1	|- ( A = x -> ( A \ y ) = ( x \ y ) )
7	6	adantl	|- ( ( ph /\ A = x ) -> ( A \ y ) = ( x \ y ) )
8		difeq2	|- ( B = y -> ( A \ B ) = ( A \ y ) )
9	8	adantl	|- ( ( ph /\ B = y ) -> ( A \ B ) = ( A \ y ) )
10		df-gdlop3	|- ~F3 = ( x e. _V , y e. _V |-> ( x \ y ) )
11	3 4 5 7 9 10	mpofvop2d	|- ( ph -> ( ~F3 ` <. A , B >. ) = ( A \ B ) )