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Theorem mopnval 18468
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 18470, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 18471. (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 5754 . . 3  |-  ( * Met `  X ) 
C_  U. ran  * Met
21sseli 3344 . 2  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
3 mopnval.1 . . 3  |-  J  =  ( MetOpen `  D )
4 fveq2 5728 . . . . . 6  |-  ( d  =  D  ->  ( ball `  d )  =  ( ball `  D
) )
54rneqd 5097 . . . . 5  |-  ( d  =  D  ->  ran  ( ball `  d )  =  ran  ( ball `  D
) )
65fveq2d 5732 . . . 4  |-  ( d  =  D  ->  ( topGen `
 ran  ( ball `  d ) )  =  ( topGen `  ran  ( ball `  D ) ) )
7 df-mopn 16698 . . . 4  |-  MetOpen  =  ( d  e.  U. ran  * Met  |->  ( topGen `  ran  ( ball `  d )
) )
8 fvex 5742 . . . 4  |-  ( topGen ` 
ran  ( ball `  D
) )  e.  _V
96, 7, 8fvmpt 5806 . . 3  |-  ( D  e.  U. ran  * Met  ->  ( MetOpen `  D
)  =  ( topGen ` 
ran  ( ball `  D
) ) )
103, 9syl5eq 2480 . 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 1652    e. wcel 1725   U.cuni 4015   ran crn 4879   ` cfv 5454   topGenctg 13665   * Metcxmt 16686   ballcbl 16688   MetOpencmopn 16691
This theorem is referenced by:  mopntopon  18469  elmopn  18472  imasf1oxms  18519  blssopn  18525  metss  18538  prdsxmslem2  18559  metcnp3  18570  metutopOLD  18612  xmetutop  18614  tgioo  18827  ismtyhmeolem  26513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-mopn 16698
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