Theorem mopnrel 15164

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 4858	. 2 |- Rel ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
2		df-mopn 14560	. . 3 |- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
3	2	releqi 4809	. 2 |- ( Rel MetOpen <-> Rel ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) ) )
4	1, 3	mpbir 146	1 |- Rel MetOpen

Syntax hints:   U.cuni 3893   |-> cmpt 4150   ran crn 4726   Rel wrel 4730   ` cfv 5326   topGenctg 13336   *Metcxmet 14549   ballcbl 14551   MetOpencmopn 14554

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-mopn 14560

This theorem is referenced by:  isxms2  15175