Theorem erdm 6779

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 6769	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1040	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3211   C_ wss 3213   `'ccnv 4750   dom cdm 4751   o. ccom 4755   Rel wrel 4756   Er wer 6766

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 1007  df-er 6769

This theorem is referenced by:  ercl  6780  erref  6789  errn  6791  erssxp  6792  erexb  6794  ereldm  6814  uniqs2  6831  iinerm  6843  th3qlem1  6873  0nnq  7681  nnnq0lem1  7763  prsrlem1  8059  gt0srpr  8065  0nsr  8066  divsfval  13558