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Metamath Proof Explorer

Theorem elmopn2

Description: A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007) (Revised by Mario Carneiro, 12-Nov-2013)

Ref Expression
Hypothesis mopnval.1 
|- J = ( MetOpen ` D )
Assertion elmopn2 
|- ( D e. ( *Met ` X ) -> ( A e. J <-> ( A C_ X /\ A. x e. A E. y e. RR+ ( x ( ball ` D ) y ) C_ A ) ) )

Proof

Step Hyp Ref Expression
1 mopnval.1 
 |-  J = ( MetOpen ` D )
2 1 elmopn 
 |-  ( D e. ( *Met ` X ) -> ( A e. J <-> ( A C_ X /\ A. x e. A E. z e. ran ( ball ` D ) ( x e. z /\ z C_ A ) ) ) )
3 ssel2 
 |-  ( ( A C_ X /\ x e. A ) -> x e. X )
4 blssex 
 |-  ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( E. z e. ran ( ball ` D ) ( x e. z /\ z C_ A ) <-> E. y e. RR+ ( x ( ball ` D ) y ) C_ A ) )
5 3 4 sylan2 
 |-  ( ( D e. ( *Met ` X ) /\ ( A C_ X /\ x e. A ) ) -> ( E. z e. ran ( ball ` D ) ( x e. z /\ z C_ A ) <-> E. y e. RR+ ( x ( ball ` D ) y ) C_ A ) )
6 5 anassrs 
 |-  ( ( ( D e. ( *Met ` X ) /\ A C_ X ) /\ x e. A ) -> ( E. z e. ran ( ball ` D ) ( x e. z /\ z C_ A ) <-> E. y e. RR+ ( x ( ball ` D ) y ) C_ A ) )
7 6 ralbidva 
 |-  ( ( D e. ( *Met ` X ) /\ A C_ X ) -> ( A. x e. A E. z e. ran ( ball ` D ) ( x e. z /\ z C_ A ) <-> A. x e. A E. y e. RR+ ( x ( ball ` D ) y ) C_ A ) )
8 7 pm5.32da 
 |-  ( D e. ( *Met ` X ) -> ( ( A C_ X /\ A. x e. A E. z e. ran ( ball ` D ) ( x e. z /\ z C_ A ) ) <-> ( A C_ X /\ A. x e. A E. y e. RR+ ( x ( ball ` D ) y ) C_ A ) ) )
9 2 8 bitrd 
 |-  ( D e. ( *Met ` X ) -> ( A e. J <-> ( A C_ X /\ A. x e. A E. y e. RR+ ( x ( ball ` D ) y ) C_ A ) ) )