Theorem elsetpreimafv

Description: An element of the class P of all preimages of function values. (Contributed by AV, 8-Mar-2024)

	Ref	Expression
Hypothesis	setpreimafvex.p	|- P = { z | E. x e. A z = ( `' F " { ( F ` x ) } ) }
Assertion	elsetpreimafv	|- ( S e. P -> E. x e. A S = ( `' F " { ( F ` x ) } ) )

Proof

Step	Hyp	Ref	Expression
1		setpreimafvex.p	|- P = { z | E. x e. A z = ( `' F " { ( F ` x ) } ) }
2	1	elsetpreimafvb	|- ( S e. P -> ( S e. P <-> E. x e. A S = ( `' F " { ( F ` x ) } ) ) )
3	2	ibi	|- ( S e. P -> E. x e. A S = ( `' F " { ( F ` x ) } ) )