Theorem opabssxp

Description: An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995)

	Ref	Expression
Assertion	opabssxp	|- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ph ) } C_ ( A X. B )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( ( x e. A /\ y e. B ) /\ ph ) -> ( x e. A /\ y e. B ) )
2	1	ssopab2i	|- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ph ) } C_ { <. x , y >. | ( x e. A /\ y e. B ) }
3		df-xp	|- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
4	2 3	sseqtrri	|- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ph ) } C_ ( A X. B )