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Theorem erdm 6300
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 6290 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
21simp2bi 959 1  |-  ( R  Er  A  ->  dom  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    u. cun 2997    C_ wss 2999   `'ccnv 4437   dom cdm 4438    o. ccom 4442   Rel wrel 4443    Er wer 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105
This theorem depends on definitions:  df-bi 115  df-3an 926  df-er 6290
This theorem is referenced by:  ercl  6301  erref  6310  errn  6312  erssxp  6313  erexb  6315  ereldm  6333  uniqs2  6350  iinerm  6362  th3qlem1  6392  0nnq  6921  nnnq0lem1  7003  prsrlem1  7286  gt0srpr  7292  0nsr  7293
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