Theorem mopnrel 14026

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 4757	. 2 ⊢ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
2		df-mopn 13536	. . 3 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
3	2	releqi 4711	. 2 ⊢ (Rel MetOpen ↔ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))))
4	1, 3	mpbir 146	1 ⊢ Rel MetOpen

Syntax hints:  ∪ cuni 3811   ↦ cmpt 4066  ran crn 4629  Rel wrel 4633  ‘cfv 5218  topGenctg 12708  ∞Metcxmet 13525  ballcbl 13527  MetOpencmopn 13530

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 4123  ax-pow 4176  ax-pr 4211

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 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067  df-mpt 4068  df-xp 4634  df-rel 4635  df-mopn 13536

This theorem is referenced by:  isxms2  14037