Theorem 1st2ndbr

Description: Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016)

	Ref	Expression
Assertion	1st2ndbr	|- ( ( Rel B /\ A e. B ) -> ( 1st ` A ) B ( 2nd ` A ) )

Proof

Step	Hyp	Ref	Expression
1		1st2nd	|- ( ( Rel B /\ A e. B ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		simpr	|- ( ( Rel B /\ A e. B ) -> A e. B )
3	1 2	eqeltrrd	|- ( ( Rel B /\ A e. B ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. B )
4		df-br	|- ( ( 1st ` A ) B ( 2nd ` A ) <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. B )
5	3 4	sylibr	|- ( ( Rel B /\ A e. B ) -> ( 1st ` A ) B ( 2nd ` A ) )