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Theorem mopnrel 13877
Description: The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
Assertion
Ref Expression
mopnrel  |-  Rel  MetOpen

Proof of Theorem mopnrel
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 mptrel 4755 . 2  |-  Rel  (
d  e.  U. ran  *Met  |->  ( topGen `  ran  ( ball `  d )
) )
2 df-mopn 13387 . . 3  |-  MetOpen  =  ( d  e.  U. ran  *Met  |->  ( topGen `  ran  ( ball `  d )
) )
32releqi 4709 . 2  |-  ( Rel  MetOpen  <->  Rel  ( d  e.  U. ran  *Met  |->  ( topGen ` 
ran  ( ball `  d
) ) ) )
41, 3mpbir 146 1  |-  Rel  MetOpen
Colors of variables: wff set class
Syntax hints:   U.cuni 3809    |-> cmpt 4064   ran crn 4627   Rel wrel 4631   ` cfv 5216   topGenctg 12702   *Metcxmet 13376   ballcbl 13378   MetOpencmopn 13381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-mpt 4066  df-xp 4632  df-rel 4633  df-mopn 13387
This theorem is referenced by:  isxms2  13888
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