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Theorem mopnval 21110
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 21112, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 21113. (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 5871 . . 3  |-  ( *Met `  X ) 
C_  U. ran  *Met
21sseli 3485 . 2  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
3 mopnval.1 . . 3  |-  J  =  ( MetOpen `  D )
4 fveq2 5848 . . . . . 6  |-  ( d  =  D  ->  ( ball `  d )  =  ( ball `  D
) )
54rneqd 5219 . . . . 5  |-  ( d  =  D  ->  ran  ( ball `  d )  =  ran  ( ball `  D
) )
65fveq2d 5852 . . . 4  |-  ( d  =  D  ->  ( topGen `
 ran  ( ball `  d ) )  =  ( topGen `  ran  ( ball `  D ) ) )
7 df-mopn 18613 . . . 4  |-  MetOpen  =  ( d  e.  U. ran  *Met  |->  ( topGen `  ran  ( ball `  d )
) )
8 fvex 5858 . . . 4  |-  ( topGen ` 
ran  ( ball `  D
) )  e.  _V
96, 7, 8fvmpt 5931 . . 3  |-  ( D  e.  U. ran  *Met  ->  ( MetOpen `  D
)  =  ( topGen ` 
ran  ( ball `  D
) ) )
103, 9syl5eq 2507 . 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 1398    e. wcel 1823   U.cuni 4235   ran crn 4989   ` cfv 5570   topGenctg 14930   *Metcxmt 18601   ballcbl 18603   MetOpencmopn 18606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-mopn 18613
This theorem is referenced by:  mopntopon  21111  elmopn  21114  imasf1oxms  21161  blssopn  21167  metss  21180  prdsxmslem2  21201  metcnp3  21212  metutopOLD  21254  xmetutop  21256  tgioo  21470  ismtyhmeolem  30543
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