MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  errel Unicode version

Theorem errel 6756
Description: An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
errel  |-  ( R  Er  A  ->  Rel  R )

Proof of Theorem errel
StepHypRef Expression
1 df-er 6747 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
21simp1bi 970 1  |-  ( R  Er  A  ->  Rel  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    u. cun 3226    C_ wss 3228   `'ccnv 4770   dom cdm 4771    o. ccom 4775   Rel wrel 4776    Er wer 6744
This theorem is referenced by:  ercl  6758  ersym  6759  ertr  6762  ercnv  6768  erssxp  6770  erth  6791  iiner  6818  frgpuplem  15180  topfneec  25615  prter3  26073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-er 6747
  Copyright terms: Public domain W3C validator