Theorem elrn2g

Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011)

	Ref	Expression
Assertion	elrn2g	|- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )

Proof

Step	Hyp	Ref	Expression
1		opeq2	|- ( y = A -> <. x , y >. = <. x , A >. )
2	1	eleq1d	|- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
3	2	exbidv	|- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
4		dfrn3	|- ran B = { y | E. x <. x , y >. e. B }
5	3 4	elab2g	|- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )