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	|- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )

Proof

Step	Hyp	Ref	Expression
1		elpreima	|- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) e. { B } ) ) )
2		fvex	|- ( F ` C ) e. _V
3	2	elsn	|- ( ( F ` C ) e. { B } <-> ( F ` C ) = B )
4	3	anbi2i	|- ( ( C e. A /\ ( F ` C ) e. { B } ) <-> ( C e. A /\ ( F ` C ) = B ) )
5	1 4	bitrdi	|- ( F Fn A -> ( C e. ( `' F " { B } ) <-> ( C e. A /\ ( F ` C ) = B ) ) )