Theorem erdm 6232

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 6222	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 955	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1285   u. cun 2982   C_ wss 2984   `'ccnv 4400   dom cdm 4401   o. ccom 4405   Rel wrel 4406   Er wer 6219

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105

This theorem depends on definitions:  df-bi 115  df-3an 922  df-er 6222

This theorem is referenced by:  ercl  6233  erref  6242  errn  6244  erssxp  6245  erexb  6247  ereldm  6265  uniqs2  6282  iinerm  6294  th3qlem1  6324  0nnq  6826  nnnq0lem1  6908  prsrlem1  7191  gt0srpr  7197  0nsr  7198