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Theorem mopnrel 12780
 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 4707 . 2 Rel (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
2 df-mopn 12330 . . 3 MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
32releqi 4662 . 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 3768   ↦ cmpt 4021  ran crn 4580  Rel wrel 4584  ‘cfv 5163  topGenctg 12305  ∞Metcxmet 12319  ballcbl 12321  MetOpencmopn 12324 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-opab 4022  df-mpt 4023  df-xp 4585  df-rel 4586  df-mopn 12330 This theorem is referenced by:  isxms2  12791
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