Theorem elpreimad

Description: Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	elpreimad.f	$$⊢\phi \to FFnA$$
	elpreimad.b	$$⊢\phi \to B\in A$$
	elpreimad.c	$$⊢\phi \to F\left(B\right)\in C$$
Assertion	elpreimad	$$⊢\phi \to B\in F^{-1}\left[C\right]$$

Proof

Step	Hyp	Ref	Expression
1		elpreimad.f	$$⊢\phi \to FFnA$$
2		elpreimad.b	$$⊢\phi \to B\in A$$
3		elpreimad.c	$$⊢\phi \to F\left(B\right)\in C$$
4		elpreima	$$⊢FFnA\to \left(B\in F^{-1}\left[C\right]\leftrightarrow \left(B\in A\wedge F\left(B\right)\in C\right)\right)$$
5	1 4	syl	$$⊢\phi \to \left(B\in F^{-1}\left[C\right]\leftrightarrow \left(B\in A\wedge F\left(B\right)\in C\right)\right)$$
6	2 3 5	mpbir2and	$$⊢\phi \to B\in F^{-1}\left[C\right]$$