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Theorem erdm 6116
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 6106 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
21simp2bi 920 1  |-  ( R  Er  A  ->  dom  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    u. cun 2915    C_ wss 2917   `'ccnv 4344   dom cdm 4345    o. ccom 4349   Rel wrel 4350    Er wer 6103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100
This theorem depends on definitions:  df-bi 110  df-3an 887  df-er 6106
This theorem is referenced by:  ercl  6117  erref  6126  errn  6128  erssxp  6129  erexb  6131  ereldm  6149  uniqs2  6166  iinerm  6178  th3qlem1  6208  0nnq  6460  nnnq0lem1  6542  prsrlem1  6825  gt0srpr  6831  0nsr  6832
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