Theorem mopnrel 15432

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 4888	. 2 ⊢ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
2		df-mopn 14821	. . 3 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
3	2	releqi 4838	. 2 ⊢ (Rel MetOpen ↔ Rel (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))))
4	1, 3	mpbir 146	1 ⊢ Rel MetOpen

Syntax hints:  ∪ cuni 3919   ↦ cmpt 4176  ran crn 4755  Rel wrel 4759  ‘cfv 5357  topGenctg 13551  ∞Metcxmet 14810  ballcbl 14812  MetOpencmopn 14815

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177  df-mpt 4178  df-xp 4760  df-rel 4761  df-mopn 14821

This theorem is referenced by:  isxms2  15443