Theorem mopnrel 15080

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 4827	. 2 ⊢ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
2		df-mopn 14476	. . 3 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
3	2	releqi 4779	. 2 ⊢ (Rel MetOpen ↔ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))))
4	1, 3	mpbir 146	1 ⊢ Rel MetOpen

Syntax hints:  ∪ cuni 3867   ↦ cmpt 4124  ran crn 4697  Rel wrel 4701  ‘cfv 5294  topGenctg 13253  ∞Metcxmet 14465  ballcbl 14467  MetOpencmopn 14470

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-opab 4125  df-mpt 4126  df-xp 4702  df-rel 4703  df-mopn 14476

This theorem is referenced by:  isxms2  15091