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Theorem mopnval 18421
Description: An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object  ( MetOpen `  D
) is the family of all open sets in the metric space determined by the metric  D. By mopntop 18423, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 18424. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
Hypothesis
Ref Expression
mopnval.1  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
mopnval  |-  ( D  e.  ( * Met `  X )  ->  J  =  ( topGen `  ran  ( ball `  D )
) )

Proof of Theorem mopnval
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 fvssunirn 5713 . . 3  |-  ( * Met `  X ) 
C_  U. ran  * Met
21sseli 3304 . 2  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
3 mopnval.1 . . 3  |-  J  =  ( MetOpen `  D )
4 fveq2 5687 . . . . . 6  |-  ( d  =  D  ->  ( ball `  d )  =  ( ball `  D
) )
54rneqd 5056 . . . . 5  |-  ( d  =  D  ->  ran  ( ball `  d )  =  ran  ( ball `  D
) )
65fveq2d 5691 . . . 4  |-  ( d  =  D  ->  ( topGen `
 ran  ( ball `  d ) )  =  ( topGen `  ran  ( ball `  D ) ) )
7 df-mopn 16653 . . . 4  |-  MetOpen  =  ( d  e.  U. ran  * Met  |->  ( topGen `  ran  ( ball `  d )
) )
8 fvex 5701 . . . 4  |-  ( topGen ` 
ran  ( ball `  D
) )  e.  _V
96, 7, 8fvmpt 5765 . . 3  |-  ( D  e.  U. ran  * Met  ->  ( MetOpen `  D
)  =  ( topGen ` 
ran  ( ball `  D
) ) )
103, 9syl5eq 2448 . 2  |-  ( D  e.  U. ran  * Met  ->  J  =  (
topGen `  ran  ( ball `  D ) ) )
112, 10syl 16 1  |-  ( D  e.  ( * Met `  X )  ->  J  =  ( topGen `  ran  ( ball `  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   U.cuni 3975   ran crn 4838   ` cfv 5413   topGenctg 13620   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646
This theorem is referenced by:  mopntopon  18422  elmopn  18425  imasf1oxms  18472  blssopn  18478  metss  18491  prdsxmslem2  18512  metcnp3  18523  metutopOLD  18565  xmetutop  18567  tgioo  18780  ismtyhmeolem  26403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fv 5421  df-mopn 16653
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