Theorem erdm 6776

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 6766	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1040	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3208   C_ wss 3210   `'ccnv 4747   dom cdm 4748   o. ccom 4752   Rel wrel 4753   Er wer 6763

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 1007  df-er 6766

This theorem is referenced by:  ercl  6777  erref  6786  errn  6788  erssxp  6789  erexb  6791  ereldm  6811  uniqs2  6828  iinerm  6840  th3qlem1  6870  0nnq  7675  nnnq0lem1  7757  prsrlem1  8053  gt0srpr  8059  0nsr  8060  divsfval  13530