Theorem erdm 6690

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 6680	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1037	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1395   u. cun 3195   C_ wss 3197   `'ccnv 4718   dom cdm 4719   o. ccom 4723   Rel wrel 4724   Er wer 6677

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 1004  df-er 6680

This theorem is referenced by:  ercl  6691  erref  6700  errn  6702  erssxp  6703  erexb  6705  ereldm  6725  uniqs2  6742  iinerm  6754  th3qlem1  6784  0nnq  7551  nnnq0lem1  7633  prsrlem1  7929  gt0srpr  7935  0nsr  7936  divsfval  13361