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Definition df-prstc 49152
Description: Definition of the function converting a preordered set to a category. Justified by prsthinc 49111.

This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism (thincciso 49102), by prstcnid 49155, prstchom 49166, and prstcthin 49165. Other important properties include prstcbas 49156, prstcleval 49157, prstcle 49159, prstcocval 49160, prstcoc 49162, prstchom2 49167, and prstcprs 49164. Use those instead.

Note that the defining property prstchom 49166 is equivalent to prstchom2 49167 given prstcthin 49165. See thincn0eu 49080 for justification.

"ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)

Assertion
Ref Expression
df-prstc ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

Detailed syntax breakdown of Definition df-prstc
StepHypRef Expression
1 cprstc 49151 . 2 class ProsetToCat
2 vk . . 3 setvar 𝑘
3 cproset 18338 . . 3 class Proset
42cv 1539 . . . . 5 class 𝑘
5 cnx 17230 . . . . . . 7 class ndx
6 chom 17308 . . . . . . 7 class Hom
75, 6cfv 6561 . . . . . 6 class (Hom ‘ndx)
8 cple 17304 . . . . . . . 8 class le
94, 8cfv 6561 . . . . . . 7 class (le‘𝑘)
10 c1o 8499 . . . . . . . 8 class 1o
1110csn 4626 . . . . . . 7 class {1o}
129, 11cxp 5683 . . . . . 6 class ((le‘𝑘) × {1o})
137, 12cop 4632 . . . . 5 class ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩
14 csts 17200 . . . . 5 class sSet
154, 13, 14co 7431 . . . 4 class (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩)
16 cco 17309 . . . . . 6 class comp
175, 16cfv 6561 . . . . 5 class (comp‘ndx)
18 c0 4333 . . . . 5 class
1917, 18cop 4632 . . . 4 class ⟨(comp‘ndx), ∅⟩
2015, 19, 14co 7431 . . 3 class ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)
212, 3, 20cmpt 5225 . 2 class (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
221, 21wceq 1540 1 wff ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Colors of variables: wff setvar class
This definition is referenced by:  prstcval  49153
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