Theorem elsetpreimafvrab

Description: An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024)

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

Proof

Step	Hyp	Ref	Expression
1		setpreimafvex.p	|- P = { z | E. x e. A z = ( `' F " { ( F ` x ) } ) }
2	1	elsetpreimafvbi	|- ( ( F Fn A /\ S e. P /\ X e. S ) -> ( y e. S <-> ( y e. A /\ ( F ` y ) = ( F ` X ) ) ) )
3		fveqeq2	|- ( x = y -> ( ( F ` x ) = ( F ` X ) <-> ( F ` y ) = ( F ` X ) ) )
4	3	elrab	|- ( y e. { x e. A | ( F ` x ) = ( F ` X ) } <-> ( y e. A /\ ( F ` y ) = ( F ` X ) ) )
5	2 4	bitr4di	|- ( ( F Fn A /\ S e. P /\ X e. S ) -> ( y e. S <-> y e. { x e. A | ( F ` x ) = ( F ` X ) } ) )
6	5	eqrdv	|- ( ( F Fn A /\ S e. P /\ X e. S ) -> S = { x e. A | ( F ` x ) = ( F ` X ) } )