Theorem mopnrel 12649

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.

Step	Hyp	Ref	Expression
1		mptrel 4675	. 2 ⊢ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
2		df-mopn 12199	. . 3 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
3	2	releqi 4630	. 2 ⊢ (Rel MetOpen ↔ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))))
4	1, 3	mpbir 145	1 ⊢ Rel MetOpen

Syntax hints:  ∪ cuni 3744   ↦ cmpt 3997  ran crn 4548  Rel wrel 4552  ‘cfv 5131  topGenctg 12174  ∞Metcxmet 12188  ballcbl 12190  MetOpencmopn 12193

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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-mpt 3999  df-xp 4553  df-rel 4554  df-mopn 12199

This theorem is referenced by:  isxms2  12660