Theorem resmpo

Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013)

	Ref	Expression
Assertion	resmpo	|- ( ( C C_ A /\ D C_ B ) -> ( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) = ( x e. C , y e. D |-> E ) )

Proof

Step	Hyp	Ref	Expression
1		resoprab2	|- ( ( C C_ A /\ D C_ B ) -> ( { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = E ) } |` ( C X. D ) ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ z = E ) } )
2		df-mpo	|- ( x e. A , y e. B |-> E ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = E ) }
3	2	reseq1i	|- ( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) = ( { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = E ) } |` ( C X. D ) )
4		df-mpo	|- ( x e. C , y e. D |-> E ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ z = E ) }
5	1 3 4	3eqtr4g	|- ( ( C C_ A /\ D C_ B ) -> ( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) = ( x e. C , y e. D |-> E ) )