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Definition df-prstc 48069
Description: Definition of the function converting a preordered set to a category. Justified by prsthinc 48060.

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, by prstcnid 48072, prstchom 48083, and prstcthin 48082. Other important properties include prstcbas 48073, prstcleval 48074, prstcle 48076, prstcocval 48077, prstcoc 48079, prstchom2 48084, and prstcprs 48081. Use those instead.

Note that the defining property prstchom 48083 is equivalent to prstchom2 48084 given prstcthin 48082. See thincn0eu 48038 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 48068 . 2 class ProsetToCat
2 vk . . 3 setvar 𝑘
3 cproset 18285 . . 3 class Proset
42cv 1533 . . . . 5 class 𝑘
5 cnx 17162 . . . . . . 7 class ndx
6 chom 17244 . . . . . . 7 class Hom
75, 6cfv 6548 . . . . . 6 class (Hom ‘ndx)
8 cple 17240 . . . . . . . 8 class le
94, 8cfv 6548 . . . . . . 7 class (le‘𝑘)
10 c1o 8480 . . . . . . . 8 class 1o
1110csn 4629 . . . . . . 7 class {1o}
129, 11cxp 5676 . . . . . 6 class ((le‘𝑘) × {1o})
137, 12cop 4635 . . . . 5 class ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩
14 csts 17132 . . . . 5 class sSet
154, 13, 14co 7420 . . . 4 class (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩)
16 cco 17245 . . . . . 6 class comp
175, 16cfv 6548 . . . . 5 class (comp‘ndx)
18 c0 4323 . . . . 5 class
1917, 18cop 4635 . . . 4 class ⟨(comp‘ndx), ∅⟩
2015, 19, 14co 7420 . . 3 class ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)
212, 3, 20cmpt 5231 . 2 class (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
221, 21wceq 1534 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  48070
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