Theorem erdm 6811

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

Step	Hyp	Ref	Expression
1		df-er 6801	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1044	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1402   u. cun 3218   C_ wss 3220   `'ccnv 4771   dom cdm 4772   o. ccom 4776   Rel wrel 4777   Er wer 6798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-er 6801

This theorem is referenced by:  ercl  6812  erref  6821  errn  6823  erssxp  6824  erexb  6826  ereldm  6846  uniqs2  6863  iinerm  6875  th3qlem1  6905  0nnq  7725  nnnq0lem1  7807  prsrlem1  8103  gt0srpr  8109  0nsr  8110  divsfval  13632