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Theorem erdm 6439
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 6429 . 2  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
21simp2bi 997 1  |-  ( R  Er  A  ->  dom  R  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    u. cun 3069    C_ wss 3071   `'ccnv 4538   dom cdm 4539    o. ccom 4543   Rel wrel 4544    Er wer 6426
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 964  df-er 6429
This theorem is referenced by:  ercl  6440  erref  6449  errn  6451  erssxp  6452  erexb  6454  ereldm  6472  uniqs2  6489  iinerm  6501  th3qlem1  6531  0nnq  7184  nnnq0lem1  7266  prsrlem1  7562  gt0srpr  7568  0nsr  7569
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