Theorem dfima3

Description: Alternate definition of image. Compare definition (d) of Enderton p. 44. (Contributed by NM, 14-Aug-1994) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	dfima3	|- ( A " B ) = { y | E. x ( x e. B /\ <. x , y >. e. A ) }

Proof

Step	Hyp	Ref	Expression
1		dfima2	|- ( A " B ) = { y | E. x e. B x A y }
2		df-br	|- ( x A y <-> <. x , y >. e. A )
3	2	rexbii	|- ( E. x e. B x A y <-> E. x e. B <. x , y >. e. A )
4		df-rex	|- ( E. x e. B <. x , y >. e. A <-> E. x ( x e. B /\ <. x , y >. e. A ) )
5	3 4	bitri	|- ( E. x e. B x A y <-> E. x ( x e. B /\ <. x , y >. e. A ) )
6	5	abbii	|- { y | E. x e. B x A y } = { y | E. x ( x e. B /\ <. x , y >. e. A ) }
7	1 6	eqtri	|- ( A " B ) = { y | E. x ( x e. B /\ <. x , y >. e. A ) }