Theorem elrnmptd

Description: The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	elrnmptd.f	|- F = ( x e. A |-> B )
	elrnmptd.x	|- ( ph -> E. x e. A C = B )
	elrnmptd.c	|- ( ph -> C e. V )
Assertion	elrnmptd	|- ( ph -> C e. ran F )

Proof

Step	Hyp	Ref	Expression
1		elrnmptd.f	|- F = ( x e. A |-> B )
2		elrnmptd.x	|- ( ph -> E. x e. A C = B )
3		elrnmptd.c	|- ( ph -> C e. V )
4	1	elrnmpt	|- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
5	3 4	syl	|- ( ph -> ( C e. ran F <-> E. x e. A C = B ) )
6	2 5	mpbird	|- ( ph -> C e. ran F )