Theorem mopnrel 13081

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 4732	. 2 |- Rel ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
2		df-mopn 12631	. . 3 |- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
3	2	releqi 4687	. 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 3789   |-> cmpt 4043   ran crn 4605   Rel wrel 4609   ` cfv 5188   topGenctg 12571   *Metcxmet 12620   ballcbl 12622   MetOpencmopn 12625

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-mpt 4045  df-xp 4610  df-rel 4611  df-mopn 12631

This theorem is referenced by:  isxms2  13092