Theorem fnsnfv

Description: Singleton of function value. (Contributed by NM, 22-May-1998)

	Ref	Expression
Assertion	fnsnfv	|- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )

Proof

Step	Hyp	Ref	Expression
1		eqcom	|- ( y = ( F ` B ) <-> ( F ` B ) = y )
2		fnbrfvb	|- ( ( F Fn A /\ B e. A ) -> ( ( F ` B ) = y <-> B F y ) )
3	1 2	syl5bb	|- ( ( F Fn A /\ B e. A ) -> ( y = ( F ` B ) <-> B F y ) )
4	3	abbidv	|- ( ( F Fn A /\ B e. A ) -> { y | y = ( F ` B ) } = { y | B F y } )
5		df-sn	|- { ( F ` B ) } = { y | y = ( F ` B ) }
6	5	a1i	|- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = { y | y = ( F ` B ) } )
7		fnrel	|- ( F Fn A -> Rel F )
8		relimasn	|- ( Rel F -> ( F " { B } ) = { y | B F y } )
9	7 8	syl	|- ( F Fn A -> ( F " { B } ) = { y | B F y } )
10	9	adantr	|- ( ( F Fn A /\ B e. A ) -> ( F " { B } ) = { y | B F y } )
11	4 6 10	3eqtr4d	|- ( ( F Fn A /\ B e. A ) -> { ( F ` B ) } = ( F " { B } ) )