Theorem preimafvsspwdm

Description: The class P of all preimages of function values is a subset of the power set of the domain of the function. (Contributed by AV, 5-Mar-2024)

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

Proof

Step	Hyp	Ref	Expression
1		setpreimafvex.p	|- P = { z | E. x e. A z = ( `' F " { ( F ` x ) } ) }
2	1	elsetpreimafvssdm	|- ( ( F Fn A /\ s e. P ) -> s C_ A )
3	2	ralrimiva	|- ( F Fn A -> A. s e. P s C_ A )
4		pwssb	|- ( P C_ ~P A <-> A. s e. P s C_ A )
5	3 4	sylibr	|- ( F Fn A -> P C_ ~P A )