Theorem erdm 6715

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 6705	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1039	1 |- ( R Er A -> dom R = A )

Syntax hints:   -> wi 4   = wceq 1397   u. cun 3198   C_ wss 3200   `'ccnv 4724   dom cdm 4725   o. ccom 4729   Rel wrel 4730   Er wer 6702

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 1006  df-er 6705

This theorem is referenced by:  ercl  6716  erref  6725  errn  6727  erssxp  6728  erexb  6730  ereldm  6750  uniqs2  6767  iinerm  6779  th3qlem1  6809  0nnq  7587  nnnq0lem1  7669  prsrlem1  7965  gt0srpr  7971  0nsr  7972  divsfval  13432