Theorem elrnmpog

Description: Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypothesis	rngop.1	|- F = ( x e. A , y e. B |-> C )
Assertion	elrnmpog	|- ( D e. V -> ( D e. ran F <-> E. x e. A E. y e. B D = C ) )

Proof

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