Definition df-preset 15110

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

Step	Hyp	Ref	Expression
1		cpreset 15108	. 2 class Preset
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1368	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . 11 setvar r
5	4	cv 1368	. . . . . . . . . 10 class r
6	3, 3, 5	wbr 4304	. . . . . . . . 9 wff x r x
7		vy	. . . . . . . . . . . . 13 setvar y
8	7	cv 1368	. . . . . . . . . . . 12 class y
9	3, 8, 5	wbr 4304	. . . . . . . . . . 11 wff x r y
10		vz	. . . . . . . . . . . . 13 setvar z
11	10	cv 1368	. . . . . . . . . . . 12 class z
12	8, 11, 5	wbr 4304	. . . . . . . . . . 11 wff y r z
13	9, 12	wa 369	. . . . . . . . . 10 wff ( x r y /\ y r z )
14	3, 11, 5	wbr 4304	. . . . . . . . . 10 wff x r z
15	13, 14	wi 4	. . . . . . . . 9 wff ( ( x r y /\ y r z ) -> x r z )
16	6, 15	wa 369	. . . . . . . 8 wff ( x r x /\ ( ( x r y /\ y r z ) -> x r z ) )
17		vb	. . . . . . . . 9 setvar b
18	17	cv 1368	. . . . . . . 8 class b
19	16, 10, 18	wral 2727	. . . . . . 7 wff A. z e. b ( x r x /\ ( ( x r y /\ y r z ) -> x r z ) )
20	19, 7, 18	wral 2727	. . . . . 6 wff A. y e. b A. z e. b ( x r x /\ ( ( x r y /\ y r z ) -> x r z ) )
21	20, 2, 18	wral 2727	. . . . 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
23	22	cv 1368	. . . . . 6 class f
24		cple 14257	. . . . . 6 class le
25	23, 24	cfv 5430	. . . . 5 class ( le ` f )
26	21, 4, 25	wsbc 3198	. . . 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 14186	. . . . 5 class Base
28	23, 27	cfv 5430	. . . 4 class ( Base ` f )
29	26, 17, 28	wsbc 3198	. . 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 ) )
30	29, 22	cab 2429	. 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 ) ) }
31	1, 30	wceq 1369	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 ) ) }

This definition is referenced by:  isprs  15112