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Metamath Proof Explorer

Theorem mopntopon

Description: The set of open sets of a metric space X is a topology on X . Remark in Kreyszig p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypothesis mopnval.1 
|- J = ( MetOpen ` D )
Assertion mopntopon 
|- ( D e. ( *Met ` X ) -> J e. ( TopOn ` X ) )

Proof

Step Hyp Ref Expression
1 mopnval.1 
 |-  J = ( MetOpen ` D )
2 1 mopnval 
 |-  ( D e. ( *Met ` X ) -> J = ( topGen ` ran ( ball ` D ) ) )
3 blbas 
 |-  ( D e. ( *Met ` X ) -> ran ( ball ` D ) e. TopBases )
4 tgtopon 
 |-  ( ran ( ball ` D ) e. TopBases -> ( topGen ` ran ( ball ` D ) ) e. ( TopOn ` U. ran ( ball ` D ) ) )
5 3 4 syl 
 |-  ( D e. ( *Met ` X ) -> ( topGen ` ran ( ball ` D ) ) e. ( TopOn ` U. ran ( ball ` D ) ) )
6 unirnbl 
 |-  ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )
7 6 fveq2d 
 |-  ( D e. ( *Met ` X ) -> ( TopOn ` U. ran ( ball ` D ) ) = ( TopOn ` X ) )
8 5 7 eleqtrd 
 |-  ( D e. ( *Met ` X ) -> ( topGen ` ran ( ball ` D ) ) e. ( TopOn ` X ) )
9 2 8 eqeltrd 
 |-  ( D e. ( *Met ` X ) -> J e. ( TopOn ` X ) )