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Theorem erdm 6407
Description: The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erdm  |-  ( R  Er  A  ->  dom  R  =  A )

Proof of Theorem erdm
StepHypRef Expression
1 df-er 6397 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
21simp2bi 982 1  |-  ( R  Er  A  ->  dom  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    u. cun 3039    C_ wss 3041   `'ccnv 4508   dom cdm 4509    o. ccom 4513   Rel wrel 4514    Er wer 6394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106
This theorem depends on definitions:  df-bi 116  df-3an 949  df-er 6397
This theorem is referenced by:  ercl  6408  erref  6417  errn  6419  erssxp  6420  erexb  6422  ereldm  6440  uniqs2  6457  iinerm  6469  th3qlem1  6499  0nnq  7140  nnnq0lem1  7222  prsrlem1  7518  gt0srpr  7524  0nsr  7525
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