Theorem opelf

Description: The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	opelf	|- ( ( F : A --> B /\ <. C , D >. e. F ) -> ( C e. A /\ D e. B ) )

Proof

Step	Hyp	Ref	Expression
1		fssxp	|- ( F : A --> B -> F C_ ( A X. B ) )
2	1	sseld	|- ( F : A --> B -> ( <. C , D >. e. F -> <. C , D >. e. ( A X. B ) ) )
3		opelxp	|- ( <. C , D >. e. ( A X. B ) <-> ( C e. A /\ D e. B ) )
4	2 3	imbitrdi	|- ( F : A --> B -> ( <. C , D >. e. F -> ( C e. A /\ D e. B ) ) )
5	4	imp	|- ( ( F : A --> B /\ <. C , D >. e. F ) -> ( C e. A /\ D e. B ) )