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Theorem mopnval 20140
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 20142, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 20143. (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 5817 . . 3  |-  ( *Met `  X ) 
C_  U. ran  *Met
21sseli 3455 . 2  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
3 mopnval.1 . . 3  |-  J  =  ( MetOpen `  D )
4 fveq2 5794 . . . . . 6  |-  ( d  =  D  ->  ( ball `  d )  =  ( ball `  D
) )
54rneqd 5170 . . . . 5  |-  ( d  =  D  ->  ran  ( ball `  d )  =  ran  ( ball `  D
) )
65fveq2d 5798 . . . 4  |-  ( d  =  D  ->  ( topGen `
 ran  ( ball `  d ) )  =  ( topGen `  ran  ( ball `  D ) ) )
7 df-mopn 17933 . . . 4  |-  MetOpen  =  ( d  e.  U. ran  *Met  |->  ( topGen `  ran  ( ball `  d )
) )
8 fvex 5804 . . . 4  |-  ( topGen ` 
ran  ( ball `  D
) )  e.  _V
96, 7, 8fvmpt 5878 . . 3  |-  ( D  e.  U. ran  *Met  ->  ( MetOpen `  D
)  =  ( topGen ` 
ran  ( ball `  D
) ) )
103, 9syl5eq 2505 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   U.cuni 4194   ran crn 4944   ` cfv 5521   topGenctg 14490   *Metcxmt 17921   ballcbl 17923   MetOpencmopn 17926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-iota 5484  df-fun 5523  df-fv 5529  df-mopn 17933
This theorem is referenced by:  mopntopon  20141  elmopn  20144  imasf1oxms  20191  blssopn  20197  metss  20210  prdsxmslem2  20231  metcnp3  20242  metutopOLD  20284  xmetutop  20286  tgioo  20500  ismtyhmeolem  28846
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