Metamath Proof Explorer


Theorem blelrnps

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 ) )

Proof

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 ) )