MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mopnval Structured version   Unicode version

Theorem mopnval 19972
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 19974, the open sets of a metric space form a topology 
J, whose base set is 
U. J by mopnuni 19975. (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 5710 . . 3  |-  ( *Met `  X ) 
C_  U. ran  *Met
21sseli 3349 . 2  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
3 mopnval.1 . . 3  |-  J  =  ( MetOpen `  D )
4 fveq2 5688 . . . . . 6  |-  ( d  =  D  ->  ( ball `  d )  =  ( ball `  D
) )
54rneqd 5063 . . . . 5  |-  ( d  =  D  ->  ran  ( ball `  d )  =  ran  ( ball `  D
) )
65fveq2d 5692 . . . 4  |-  ( d  =  D  ->  ( topGen `
 ran  ( ball `  d ) )  =  ( topGen `  ran  ( ball `  D ) ) )
7 df-mopn 17772 . . . 4  |-  MetOpen  =  ( d  e.  U. ran  *Met  |->  ( topGen `  ran  ( ball `  d )
) )
8 fvex 5698 . . . 4  |-  ( topGen ` 
ran  ( ball `  D
) )  e.  _V
96, 7, 8fvmpt 5771 . . 3  |-  ( D  e.  U. ran  *Met  ->  ( MetOpen `  D
)  =  ( topGen ` 
ran  ( ball `  D
) ) )
103, 9syl5eq 2485 . 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 1364    e. wcel 1761   U.cuni 4088   ran crn 4837   ` cfv 5415   topGenctg 14372   *Metcxmt 17760   ballcbl 17762   MetOpencmopn 17765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fv 5423  df-mopn 17772
This theorem is referenced by:  mopntopon  19973  elmopn  19976  imasf1oxms  20023  blssopn  20029  metss  20042  prdsxmslem2  20063  metcnp3  20074  metutopOLD  20116  xmetutop  20118  tgioo  20332  ismtyhmeolem  28628
  Copyright terms: Public domain W3C validator