Theorem erdm 6544

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 6534	. 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 3127   C_ wss 3129   `'ccnv 4625   dom cdm 4626   o. ccom 4630   Rel wrel 4631   Er wer 6531

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 6534

This theorem is referenced by:  ercl  6545  erref  6554  errn  6556  erssxp  6557  erexb  6559  ereldm  6577  uniqs2  6594  iinerm  6606  th3qlem1  6636  0nnq  7362  nnnq0lem1  7444  prsrlem1  7740  gt0srpr  7746  0nsr  7747