Theorem erdm 6447

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 6437	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 998	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1332   u. cun 3074   C_ wss 3076   `'ccnv 4546   dom cdm 4547   o. ccom 4551   Rel wrel 4552   Er wer 6434

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 965  df-er 6437

This theorem is referenced by:  ercl  6448  erref  6457  errn  6459  erssxp  6460  erexb  6462  ereldm  6480  uniqs2  6497  iinerm  6509  th3qlem1  6539  0nnq  7196  nnnq0lem1  7278  prsrlem1  7574  gt0srpr  7580  0nsr  7581