Theorem mopnrel 14609

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 4790	. 2 |- Rel ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
2		df-mopn 14043	. . 3 |- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
3	2	releqi 4742	. 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 3835   |-> cmpt 4090   ran crn 4660   Rel wrel 4664   ` cfv 5254   topGenctg 12865   *Metcxmet 14032   ballcbl 14034   MetOpencmopn 14037

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-mpt 4092  df-xp 4665  df-rel 4666  df-mopn 14043

This theorem is referenced by:  isxms2  14620