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 = w -> <. x , y >. = <. x , w >. )
2	1	eleq1d	|- ( y = w -> ( <. x , y >. e. B <-> <. x , w >. e. B ) )
3	2	exbidv	|- ( y = w -> ( E. x <. x , y >. e. B <-> E. x <. x , w >. e. B ) )
4		opeq2	|- ( w = A -> <. x , w >. = <. x , A >. )
5	4	eleq1d	|- ( w = A -> ( <. x , w >. e. B <-> <. x , A >. e. B ) )
6	5	exbidv	|- ( w = A -> ( E. x <. x , w >. e. B <-> E. x <. x , A >. e. B ) )
7		dfrn3	|- ran B = { y | E. x <. x , y >. e. B }
8	3 6 7	elab2gw	|- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )