Theorem blrn

Description: Membership in the range of the ball function. Note that ran ( ballD ) is the collection of all balls for metric D . (Contributed by NM, 31-Aug-2006) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Assertion	blrn	|- ( D e. ( *Met ` X ) -> ( A e. ran ( ball ` D ) <-> E. x e. X E. r e. RR* A = ( x ( ball ` D ) r ) ) )

Proof

Step	Hyp	Ref	Expression
1		blf	|- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2		ffn	|- ( ( ball ` D ) : ( X X. RR* ) --> ~P X -> ( ball ` D ) Fn ( X X. RR* ) )
3		ovelrn	|- ( ( ball ` D ) Fn ( X X. RR* ) -> ( A e. ran ( ball ` D ) <-> E. x e. X E. r e. RR* A = ( x ( ball ` D ) r ) ) )
4	1 2 3	3syl	|- ( D e. ( *Met ` X ) -> ( A e. ran ( ball ` D ) <-> E. x e. X E. r e. RR* A = ( x ( ball ` D ) r ) ) )