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Definition df-er 6657
Description: Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6658 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6677, ersymb 6671, and ertr 6672. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)
Assertion
Ref Expression
df-er  |-  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )

Detailed syntax breakdown of Definition df-er
StepHypRef Expression
1 cA . . 3  class  A
2 cR . . 3  class  R
31, 2wer 6654 . 2  wff  R  Er  A
42wrel 4695 . . 3  wff  Rel  R
52cdm 4690 . . . 4  class  dom  R
65, 1wceq 1625 . . 3  wff  dom  R  =  A
72ccnv 4689 . . . . 5  class  `' R
82, 2ccom 4694 . . . . 5  class  ( R  o.  R )
97, 8cun 3153 . . . 4  class  ( `' R  u.  ( R  o.  R ) )
109, 2wss 3155 . . 3  wff  ( `' R  u.  ( R  o.  R ) ) 
C_  R
114, 6, 10w3a 936 . 2  wff  ( Rel 
R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R )
123, 11wb 178 1  wff  ( R  Er  A  <->  ( Rel  R  /\  dom  R  =  A  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
Colors of variables: wff set class
This definition is referenced by:  dfer2  6658  ereq1  6664  ereq2  6665  errel  6666  erdm  6667  ersym  6669  ertr  6672  xpider  6727
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