Metamath Proof Explorer

Theorem blssm

Description: A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Assertion blssm 
|- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X )

Proof

Step Hyp Ref Expression
1 blf 
 |-  ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2 fovrn 
 |-  ( ( ( ball ` D ) : ( X X. RR* ) --> ~P X /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) e. ~P X )
3 1 2 syl3an1 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) e. ~P X )
4 3 elpwid 
 |-  ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X )