Theorem erdm 6523

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 6513	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1008	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1348   u. cun 3119   C_ wss 3121   `'ccnv 4610   dom cdm 4611   o. ccom 4615   Rel wrel 4616   Er wer 6510

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 975  df-er 6513

This theorem is referenced by:  ercl  6524  erref  6533  errn  6535  erssxp  6536  erexb  6538  ereldm  6556  uniqs2  6573  iinerm  6585  th3qlem1  6615  0nnq  7326  nnnq0lem1  7408  prsrlem1  7704  gt0srpr  7710  0nsr  7711