Theorem bj-imdirval

Description: Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023)

	Ref	Expression
Hypotheses	bj-imdirval.1	|- ( ph -> A e. U )
	bj-imdirval.2	|- ( ph -> B e. V )
Assertion	bj-imdirval	|- ( ph -> ( A ~P_* B ) = ( r e. ~P ( A X. B ) |-> { <. x , y >. | ( ( x C_ A /\ y C_ B ) /\ ( r " x ) = y ) } ) )

Proof

Step	Hyp	Ref	Expression
1		bj-imdirval.1	|- ( ph -> A e. U )
2		bj-imdirval.2	|- ( ph -> B e. V )
3		df-imdir	|- ~P_* = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> { <. x , y >. | ( ( x C_ a /\ y C_ b ) /\ ( r " x ) = y ) } ) )
4	1 2 3	bj-imdirvallem	|- ( ph -> ( A ~P_* B ) = ( r e. ~P ( A X. B ) |-> { <. x , y >. | ( ( x C_ A /\ y C_ B ) /\ ( r " x ) = y ) } ) )