Theorem erdm 6632

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 6622	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1016	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1373   u. cun 3164   C_ wss 3166   `'ccnv 4675   dom cdm 4676   o. ccom 4680   Rel wrel 4681   Er wer 6619

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 983  df-er 6622

This theorem is referenced by:  ercl  6633  erref  6642  errn  6644  erssxp  6645  erexb  6647  ereldm  6667  uniqs2  6684  iinerm  6696  th3qlem1  6726  0nnq  7479  nnnq0lem1  7561  prsrlem1  7857  gt0srpr  7863  0nsr  7864  divsfval  13193