Theorem elrn2

Description: Membership in a range. (Contributed by NM, 10-Jul-1994)

	Ref	Expression
Hypothesis	elrn.1	|- A e. _V
Assertion	elrn2	|- ( A e. ran B <-> E. x <. x , A >. e. B )

Proof

Step	Hyp	Ref	Expression
1		elrn.1	|- A e. _V
2		opeq2	|- ( y = A -> <. x , y >. = <. x , A >. )
3	2	eleq1d	|- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
4	3	exbidv	|- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
5		dfrn3	|- ran B = { y | E. x <. x , y >. e. B }
6	1 4 5	elab2	|- ( A e. ran B <-> E. x <. x , A >. e. B )