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Theorem erdm 6052
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 6042 . 2 (𝑅 Er A ↔ (Rel 𝑅 dom 𝑅 = A (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
21simp2bi 919 1 (𝑅 Er A → dom 𝑅 = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cun 2909  wss 2911  ccnv 4287  dom cdm 4288  ccom 4292  Rel wrel 4293   Er wer 6039
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 886  df-er 6042
This theorem is referenced by:  ercl  6053  erref  6062  errn  6064  erssxp  6065  erexb  6067  ereldm  6085  uniqs2  6102  iinerm  6114  th3qlem1  6144  0nnq  6348  nnnq0lem1  6428  prsrlem1  6650  gt0srpr  6656  0nsr  6657
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