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	$$⊢P=\left\{z|\exists x\in Az=F^{-1}\left[\left\{F\left(x\right)\right\}\right]\right\}$$
Assertion	elsetpreimafv	$$⊢S\in P\to \exists x\in AS=F^{-1}\left[\left\{F\left(x\right)\right\}\right]$$

Proof

Step	Hyp	Ref	Expression
1		setpreimafvex.p	$$⊢P=\left\{z|\exists x\in Az=F^{-1}\left[\left\{F\left(x\right)\right\}\right]\right\}$$
2	1	elsetpreimafvb	$$⊢S\in P\to \left(S\in P\leftrightarrow \exists x\in AS=F^{-1}\left[\left\{F\left(x\right)\right\}\right]\right)$$
3	2	ibi	$$⊢S\in P\to \exists x\in AS=F^{-1}\left[\left\{F\left(x\right)\right\}\right]$$