Theorem mopnrel 12424

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 4625	. 2 |- Rel ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
2		df-mopn 11997	. . 3 |- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
3	2	releqi 4580	. 2 |- ( Rel MetOpen <-> Rel ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) ) )
4	1, 3	mpbir 145	1 |- Rel MetOpen

Syntax hints:   U.cuni 3700   |-> cmpt 3947   ran crn 4498   Rel wrel 4502   ` cfv 5079   topGenctg 11972   *Metcxmet 11986   ballcbl 11988   MetOpencmopn 11991

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-opab 3948  df-mpt 3949  df-xp 4503  df-rel 4504  df-mopn 11997

This theorem is referenced by:  isxms2  12435