Theorem fniniseg

Description: Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014) (Proof shortened by Mario Carneiro , 28-Apr-2015)

	Ref	Expression
Assertion	fniniseg	$$⊢FFnA\to \left(C\in F^{-1}\left[\left\{B\right\}\right]\leftrightarrow \left(C\in A\wedge F\left(C\right)=B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elpreima	$$⊢FFnA\to \left(C\in F^{-1}\left[\left\{B\right\}\right]\leftrightarrow \left(C\in A\wedge F\left(C\right)\in \left\{B\right\}\right)\right)$$
2		fvex	$$⊢F\left(C\right)\in \mathrm{V}$$
3	2	elsn	$$⊢F\left(C\right)\in \left\{B\right\}\leftrightarrow F\left(C\right)=B$$
4	3	anbi2i	$$⊢\left(C\in A\wedge F\left(C\right)\in \left\{B\right\}\right)\leftrightarrow \left(C\in A\wedge F\left(C\right)=B\right)$$
5	1 4	bitrdi	$$⊢FFnA\to \left(C\in F^{-1}\left[\left\{B\right\}\right]\leftrightarrow \left(C\in A\wedge F\left(C\right)=B\right)\right)$$