Theorem elrnressn

Description: Element of the range of a restriction to a singleton. (Contributed by Peter Mazsa, 12-Jun-2024)

	Ref	Expression
Assertion	elrnressn	|- ( ( A e. V /\ B e. W ) -> ( B e. ran ( R |` { A } ) <-> A R B ) )

Proof

Step	Hyp	Ref	Expression
1		elrnres	|- ( B e. W -> ( B e. ran ( R |` { A } ) <-> E. x e. { A } x R B ) )
2		breq1	|- ( x = A -> ( x R B <-> A R B ) )
3	2	rexsng	|- ( A e. V -> ( E. x e. { A } x R B <-> A R B ) )
4	1 3	sylan9bbr	|- ( ( A e. V /\ B e. W ) -> ( B e. ran ( R |` { A } ) <-> A R B ) )