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Definition df-preset 15684
Description: Define the class of preordered sets (presets). A preset is a set equipped with a transitive and reflexive relation.

Preorders are a natural generalization of order for sets where there is a well-defined ordering, but it in some sense "fails to capture the whole story", in that there may be pairs of elements which are indistinguishable under the order. Two elements which are not equal but are less-or-equal to each other behave the same under all order operations and may be thought of as "tied".

A preorder can naturally be strengthened by requiring that there are no ties, resulting in a partial order, or by stating that all comparable pairs of elements are tied, resulting in an equivalence relation. Every preorder naturally factors into these two types; the tied relation on a preorder is an equivalence relation and the quotient under that relation is a partial order. (Contributed by FL, 17-Nov-2014.) (Revised by Stefan O'Rear, 31-Jan-2015.)

Assertion
Ref Expression
df-preset  |-  Preset  =  {
f  |  [. ( Base `  f )  / 
b ]. [. ( le
`  f )  / 
r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) ) }
Distinct variable group:    f, b, r, x, y, z

Detailed syntax breakdown of Definition df-preset
StepHypRef Expression
1 cpreset 15682 . 2  class  Preset
2 vx . . . . . . . . . . 11  setvar  x
32cv 1394 . . . . . . . . . 10  class  x
4 vr . . . . . . . . . . 11  setvar  r
54cv 1394 . . . . . . . . . 10  class  r
63, 3, 5wbr 4456 . . . . . . . . 9  wff  x r x
7 vy . . . . . . . . . . . . 13  setvar  y
87cv 1394 . . . . . . . . . . . 12  class  y
93, 8, 5wbr 4456 . . . . . . . . . . 11  wff  x r y
10 vz . . . . . . . . . . . . 13  setvar  z
1110cv 1394 . . . . . . . . . . . 12  class  z
128, 11, 5wbr 4456 . . . . . . . . . . 11  wff  y r z
139, 12wa 369 . . . . . . . . . 10  wff  ( x r y  /\  y
r z )
143, 11, 5wbr 4456 . . . . . . . . . 10  wff  x r z
1513, 14wi 4 . . . . . . . . 9  wff  ( ( x r y  /\  y r z )  ->  x r z )
166, 15wa 369 . . . . . . . 8  wff  ( x r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
17 vb . . . . . . . . 9  setvar  b
1817cv 1394 . . . . . . . 8  class  b
1916, 10, 18wral 2807 . . . . . . 7  wff  A. z  e.  b  ( x
r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
2019, 7, 18wral 2807 . . . . . 6  wff  A. y  e.  b  A. z  e.  b  ( x
r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
2120, 2, 18wral 2807 . . . . 5  wff  A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x
r x  /\  (
( x r y  /\  y r z )  ->  x r
z ) )
22 vf . . . . . . 7  setvar  f
2322cv 1394 . . . . . 6  class  f
24 cple 14719 . . . . . 6  class  le
2523, 24cfv 5594 . . . . 5  class  ( le
`  f )
2621, 4, 25wsbc 3327 . . . 4  wff  [. ( le `  f )  / 
r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) )
27 cbs 14644 . . . . 5  class  Base
2823, 27cfv 5594 . . . 4  class  ( Base `  f )
2926, 17, 28wsbc 3327 . . 3  wff  [. ( Base `  f )  / 
b ]. [. ( le
`  f )  / 
r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) )
3029, 22cab 2442 . 2  class  { f  |  [. ( Base `  f )  /  b ]. [. ( le `  f )  /  r ]. A. x  e.  b 
A. y  e.  b 
A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y
r z )  ->  x r z ) ) }
311, 30wceq 1395 1  wff  Preset  =  {
f  |  [. ( Base `  f )  / 
b ]. [. ( le
`  f )  / 
r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) ) }
Colors of variables: wff setvar class
This definition is referenced by:  isprs  15686
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