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Theorem erdm 6232
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 6222 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
21simp2bi 955 1  |-  ( R  Er  A  ->  dom  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    u. cun 2982    C_ wss 2984   `'ccnv 4400   dom cdm 4401    o. ccom 4405   Rel wrel 4406    Er wer 6219
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 922  df-er 6222
This theorem is referenced by:  ercl  6233  erref  6242  errn  6244  erssxp  6245  erexb  6247  ereldm  6265  uniqs2  6282  iinerm  6294  th3qlem1  6324  0nnq  6826  nnnq0lem1  6908  prsrlem1  7191  gt0srpr  7197  0nsr  7198
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