Description: A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013) (Revised by Thierry Arnoux, 11-Mar-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | blelrnps | |- ( ( D e. ( PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) e. ran ( ball ` D ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blfps | |- ( D e. ( PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X ) |
|
2 | 1 | ffnd | |- ( D e. ( PsMet ` X ) -> ( ball ` D ) Fn ( X X. RR* ) ) |
3 | fnovrn | |- ( ( ( ball ` D ) Fn ( X X. RR* ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) e. ran ( ball ` D ) ) |
|
4 | 2 3 | syl3an1 | |- ( ( D e. ( PsMet ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) e. ran ( ball ` D ) ) |