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Definition df-preset 16922
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]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
Distinct variable group:   𝑓,𝑏,𝑟,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-preset
StepHypRef Expression
1 cpreset 16920 . 2 class Preset
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1481 . . . . . . . . . 10 class 𝑥
4 vr . . . . . . . . . . 11 setvar 𝑟
54cv 1481 . . . . . . . . . 10 class 𝑟
63, 3, 5wbr 4651 . . . . . . . . 9 wff 𝑥𝑟𝑥
7 vy . . . . . . . . . . . . 13 setvar 𝑦
87cv 1481 . . . . . . . . . . . 12 class 𝑦
93, 8, 5wbr 4651 . . . . . . . . . . 11 wff 𝑥𝑟𝑦
10 vz . . . . . . . . . . . . 13 setvar 𝑧
1110cv 1481 . . . . . . . . . . . 12 class 𝑧
128, 11, 5wbr 4651 . . . . . . . . . . 11 wff 𝑦𝑟𝑧
139, 12wa 384 . . . . . . . . . 10 wff (𝑥𝑟𝑦𝑦𝑟𝑧)
143, 11, 5wbr 4651 . . . . . . . . . 10 wff 𝑥𝑟𝑧
1513, 14wi 4 . . . . . . . . 9 wff ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)
166, 15wa 384 . . . . . . . 8 wff (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))
17 vb . . . . . . . . 9 setvar 𝑏
1817cv 1481 . . . . . . . 8 class 𝑏
1916, 10, 18wral 2911 . . . . . . 7 wff 𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))
2019, 7, 18wral 2911 . . . . . 6 wff 𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))
2120, 2, 18wral 2911 . . . . 5 wff 𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))
22 vf . . . . . . 7 setvar 𝑓
2322cv 1481 . . . . . 6 class 𝑓
24 cple 15942 . . . . . 6 class le
2523, 24cfv 5886 . . . . 5 class (le‘𝑓)
2621, 4, 25wsbc 3433 . . . 4 wff [(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))
27 cbs 15851 . . . . 5 class Base
2823, 27cfv 5886 . . . 4 class (Base‘𝑓)
2926, 17, 28wsbc 3433 . . 3 wff [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))
3029, 22cab 2607 . 2 class {𝑓[(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
311, 30wceq 1482 1 wff Preset = {𝑓[(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
Colors of variables: wff setvar class
This definition is referenced by:  isprs  16924
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