Theorem mopnrel 12610

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 4667	. 2 ⊢ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
2		df-mopn 12160	. . 3 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
3	2	releqi 4622	. 2 ⊢ (Rel MetOpen ↔ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))))
4	1, 3	mpbir 145	1 ⊢ Rel MetOpen

Syntax hints:  ∪ cuni 3736   ↦ cmpt 3989  ran crn 4540  Rel wrel 4544  ‘cfv 5123  topGenctg 12135  ∞Metcxmet 12149  ballcbl 12151  MetOpencmopn 12154

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-mpt 3991  df-xp 4545  df-rel 4546  df-mopn 12160

This theorem is referenced by:  isxms2  12621