Theorem oprab4

Description: Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010)

	Ref	Expression
Assertion	oprab4	|- { <. <. x , y >. , z >. | ( <. x , y >. e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }

Proof

Step	Hyp	Ref	Expression
1		opelxp	|- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
2	1	anbi1i	|- ( ( <. x , y >. e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ph ) )
3	2	oprabbii	|- { <. <. x , y >. , z >. | ( <. x , y >. e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }