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Theorem mopnrel 12505
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 4635 . 2 Rel (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
2 df-mopn 12055 . . 3 MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
32releqi 4590 . 2 (Rel MetOpen ↔ Rel (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑))))
41, 3mpbir 145 1 Rel MetOpen
Colors of variables: wff set class
Syntax hints:   cuni 3704  cmpt 3957  ran crn 4508  Rel wrel 4512  cfv 5091  topGenctg 12030  ∞Metcxmet 12044  ballcbl 12046  MetOpencmopn 12049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-mpt 3959  df-xp 4513  df-rel 4514  df-mopn 12055
This theorem is referenced by:  isxms2  12516
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