Theorem erdm 6822

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 6812	. 2 |- ( R Er A <-> ( Rel R /\ dom R = A /\ ( `' R u. ( R o. R ) ) C_ R ) )
2	1	simp2bi 1044	1 |- ( R Er A -> dom R = A )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   u. cun 3218   C_ wss 3220   `'ccnv 4775   dom cdm 4776   o. ccom 4780   Rel wrel 4781   Er wer 6809

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This proof depends on definitions:  df-bi 117  df-3an 1011  df-er 6812

This theorem is used by:  ercl  6823  erref  6832  errn  6834  erssxp  6835  erexb  6837  ereldm  6857  uniqs2  6874  iinerm  6886  th3qlem1  6916  0nnq  7738  nnnq0lem1  7820  prsrlem1  8116  gt0srpr  8122  0nsr  8123  divsfval  13708