Theorem erdm 6653

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 6643	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1016	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1373   u. cun 3172   C_ wss 3174   `'ccnv 4692   dom cdm 4693   o. ccom 4697   Rel wrel 4698   Er wer 6640

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 983  df-er 6643

This theorem is referenced by:  ercl  6654  erref  6663  errn  6665  erssxp  6666  erexb  6668  ereldm  6688  uniqs2  6705  iinerm  6717  th3qlem1  6747  0nnq  7512  nnnq0lem1  7594  prsrlem1  7890  gt0srpr  7896  0nsr  7897  divsfval  13275