Theorem elrnmpt

Description: The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015)

	Ref	Expression
Hypothesis	rnmpt.1	|- F = ( x e. A |-> B )
Assertion	elrnmpt	|- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Proof

Step	Hyp	Ref	Expression
1		rnmpt.1	|- F = ( x e. A |-> B )
2		eqeq1	|- ( y = C -> ( y = B <-> C = B ) )
3	2	rexbidv	|- ( y = C -> ( E. x e. A y = B <-> E. x e. A C = B ) )
4	1	rnmpt	|- ran F = { y | E. x e. A y = B }
5	3 4	elab2g	|- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )