ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  errel Unicode version

Theorem errel 6563
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 6554 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
21simp1bi 1014 1  |-  ( R  Er  A  ->  Rel  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    u. cun 3142    C_ wss 3144   `'ccnv 4640   dom cdm 4641    o. ccom 4645   Rel wrel 4646    Er wer 6551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106
This theorem depends on definitions:  df-bi 117  df-3an 982  df-er 6554
This theorem is referenced by:  ercl  6565  ersym  6566  ertr  6569  ercnv  6575  erssxp  6577  erth  6600  iinerm  6628  eqg0el  13161
  Copyright terms: Public domain W3C validator