Theorem mopnrel 15542

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 4908	. 2 |- Rel ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
2		df-mopn 14884	. . 3 |- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
3	2	releqi 4858	. 2 |- ( Rel MetOpen <-> Rel ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) ) )
4	1, 3	mpbir 146	1 |- Rel MetOpen

This proof depends on syntax axioms:   U.cuni 3935   |-> cmpt 4192   ran crn 4775   Rel wrel 4779   ` cfv 5377   topGenctg 13608   *Metcxmet 14873   ballcbl 14875   MetOpencmopn 14878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-mpt 4194  df-xp 4780  df-rel 4781  df-mopn 14884

This theorem is used by:  isxms2  15553