Theorem erdm 6545

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 6535	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1013	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1353   u. cun 3128   C_ wss 3130   `'ccnv 4626   dom cdm 4627   o. ccom 4631   Rel wrel 4632   Er wer 6532

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 980  df-er 6535

This theorem is referenced by:  ercl  6546  erref  6555  errn  6557  erssxp  6558  erexb  6560  ereldm  6578  uniqs2  6595  iinerm  6607  th3qlem1  6637  0nnq  7363  nnnq0lem1  7445  prsrlem1  7741  gt0srpr  7747  0nsr  7748