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Theorem erdm 6544
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 6534 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
21simp2bi 1013 1  |-  ( R  Er  A  ->  dom  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    u. cun 3127    C_ wss 3129   `'ccnv 4625   dom cdm 4626    o. ccom 4630   Rel wrel 4631    Er wer 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107
This theorem depends on definitions:  df-bi 117  df-3an 980  df-er 6534
This theorem is referenced by:  ercl  6545  erref  6554  errn  6556  erssxp  6557  erexb  6559  ereldm  6577  uniqs2  6594  iinerm  6606  th3qlem1  6636  0nnq  7362  nnnq0lem1  7444  prsrlem1  7740  gt0srpr  7746  0nsr  7747
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