P. 481 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48001-48100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ipoglbdm 48001*	The domain of the GLB of the inclusion poset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    ⇒   ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑇 ∈ 𝐹))
Theorem	ipoglb 48002*	The GLB of the inclusion poset. (hypotheses "ipolub.s" and "ipoglb.t" could be eliminated with 𝑆 ∈ dom 𝐺.) Could be significantly shortened if posglbdg 18407 is in quantified form. mrelatglb 18552 could potentially be shortened using this. See mrelatglbALT 48007. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    &   ⊢ (𝜑 → 𝑇 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇)
Theorem	ipolub0 48003	The LUB of the empty set is the intersection of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → ∩ 𝐹 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑈‘∅) = ∩ 𝐹)
Theorem	ipolub00 48004	The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐼))    &   ⊢ (𝜑 → ∅ ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝑈‘∅) = ∅)
Theorem	ipoglb0 48005	The GLB of the empty set is the union of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐼))    &   ⊢ (𝜑 → ∪ 𝐹 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐹)
Theorem	mrelatlubALT 48006	Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐹 = (mrCls‘𝐶)    &   ⊢ 𝐿 = (lub‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))
Theorem	mrelatglbALT 48007	Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈)
Theorem	mreclat 48008	A Moore space is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐶)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)
Theorem	topclat 48009	A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat)
Theorem	toplatglb0 48010	The empty intersection in a topology is realized by the base set. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐽)
Theorem	toplatlub 48011	Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐽)    &   ⊢ 𝑈 = (lub‘𝐼)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = ∪ 𝑆)
Theorem	toplatglb 48012	Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐽)    &   ⊢ 𝐺 = (glb‘𝐼)    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆))
Theorem	toplatjoin 48013	Joins in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐽)    &   ⊢ ∨ = (join‘𝐼)    ⇒   ⊢ (𝜑 → (𝐴 ∨ 𝐵) = (𝐴 ∪ 𝐵))
Theorem	toplatmeet 48014	Meets in a topology are realized by intersections. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐽)    &   ⊢ ∧ = (meet‘𝐼)    ⇒   ⊢ (𝜑 → (𝐴 ∧ 𝐵) = (𝐴 ∩ 𝐵))
Theorem	topdlat 48015	A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐼 = (toInc‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat)
21.46.12  Categories
21.46.12.1  Categories
Theorem	catprslem 48016*	Lemma for catprs 48017. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
Theorem	catprs 48017*	A preorder can be extracted from a category. See catprs2 48018 for more details. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))
Theorem	catprs2 48018*	A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 48019 and catprsc2 48020 for constructions satisfying the hypothesis "catprs.1". See catprs 48017 for a more primitive version. See prsthinc 48060 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ∈ Proset )
Theorem	catprsc 48019*	A construction of the preorder induced by a category. See catprs2 48018 for details. See also catprsc2 48020 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)})    ⇒   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
Theorem	catprsc2 48020*	An alternate construction of the preorder induced by a category. See catprs2 48018 for details. See also catprsc 48019 for a different construction. The two constructions are different because df-cat 17648 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅})    ⇒   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
Theorem	endmndlem 48021	A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 48090 for converting a monoid to a category. Lemma for bj-endmnd 36797. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀))    &   ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘𝑀))    ⇒   ⊢ (𝜑 → 𝑀 ∈ Mnd)
21.46.12.2  Monomorphisms and epimorphisms
Theorem	idmon 48022	An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑀 = (Mono‘𝐶)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋))
Theorem	idepi 48023	An identity arrow, or an identity morphism, is an epimorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐸 = (Epi‘𝐶)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐸𝑋))
21.46.12.3  Functors
Theorem	funcf2lem 48024*	A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
21.46.13  Examples of categories
21.46.13.1  Thin categories
Syntax	cthinc 48025	Extend class notation with the class of thin categories.
class ThinCat
Definition	df-thinc 48026*	Definition of the class of thin categories, or posetal categories, whose hom-sets each contain at most one morphism. Example 3.26(2) of [Adamek] p. 33. "ThinCat" was taken instead of "PosCat" because the latter might mean the category of posets. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
Theorem	isthinc 48027*	The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
Theorem	isthinc2 48028*	A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1o. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o))
Theorem	isthinc3 48029*	A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
Theorem	thincc 48030	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
Theorem	thinccd 48031	A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	thincssc 48032	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat ⊆ Cat
Theorem	isthincd2lem1 48033*	Lemma for isthincd2 48044 and thincmo2 48034. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	thincmo2 48034	Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	thincmo 48035*	There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincmoALT 48036*	Alternate proof for thincmo 48035. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincmod 48037*	At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Theorem	thincn0eu 48038*	In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	thincid 48039	In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))    ⇒   ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
Theorem	thincmon 48040	In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 48102. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑀 = (Mono‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
Theorem	thincepi 48041	In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 48103. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐸 = (Epi‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
Theorem	isthincd2lem2 48042*	Lemma for isthincd2 48044. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))
Theorem	isthincd 48043*	The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	isthincd2 48044*	The predicate "𝐶 is a thin category" without knowing 𝐶 is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))))    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦))    &   ⊢ ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 )))
Theorem	oppcthin 48045	The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat)
Theorem	subthinc 48046	A subcategory of a thin category is thin. (Contributed by Zhi Wang, 30-Sep-2024.)
⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐷 ∈ ThinCat)
Theorem	functhinclem1 48047*	Lemma for functhinc 48051. Given the object part, there is only one possible morphism part such that the mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    ⇒   ⊢ (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ↔ 𝐺 = 𝐾))
Theorem	functhinclem2 48048*	Lemma for functhinc 48051. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))    ⇒   ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
Theorem	functhinclem3 48049*	Lemma for functhinc 48051. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))    &   ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))    &   ⊢ (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	functhinclem4 48050*	Lemma for functhinc 48051. Other requirements on the morphism part are automatically satisfied. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ 𝐼 = (Id‘𝐸)    &   ⊢ · = (comp‘𝐷)    &   ⊢ 𝑂 = (comp‘𝐸)    ⇒   ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘( 1 ‘𝑎)) = (𝐼‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑚 ∈ (𝑎𝐻𝑏)∀𝑛 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑛(⟨𝑎, 𝑏⟩ · 𝑐)𝑚)) = (((𝑏𝐺𝑐)‘𝑛)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩𝑂(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑚))))
Theorem	functhinc 48051*	A functor to a thin category is determined entirely by the object part. The hypothesis "functhinc.1" is related to a monotone function if preorders induced by the categories are considered (catprs2 48018). (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    ⇒   ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾))
Theorem	fullthinc 48052*	A functor to a thin category is full iff empty hom-sets are mapped to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))
Theorem	fullthinc2 48053	A full functor to a thin category maps empty hom-sets to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
Theorem	thincfth 48054	A functor from a thin category is faithful. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
Theorem	thincciso 48055*	Two thin categories are isomorphic iff the induced preorders are order-isomorphic. Example 3.26(2) of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Oct-2024.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑅 = (Base‘𝑋)    &   ⊢ 𝑆 = (Base‘𝑌)    &   ⊢ 𝐻 = (Hom ‘𝑋)    &   ⊢ 𝐽 = (Hom ‘𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑌 ∈ ThinCat)    ⇒   ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
Theorem	0thincg 48056	Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat)
Theorem	0thinc 48057	The empty category (see 0cat 17669) is thin. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ∅ ∈ ThinCat
Theorem	indthinc 48058*	An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are ∅. This is a special case of prsthinc 48060, where ≤ = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))    &   ⊢ (𝜑 → ∅ = (comp‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
Theorem	indthincALT 48059*	An alternate proof for indthinc 48058 assuming more axioms including ax-pow 5365 and ax-un 7740. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))    &   ⊢ (𝜑 → ∅ = (comp‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
Theorem	prsthinc 48060*	Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 48017 and catprs2 48018 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))    &   ⊢ (𝜑 → ∅ = (comp‘𝐶))    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
Theorem	setcthin 48061*	A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (SetCat‘𝑈))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	setc2othin 48062	The category (SetCat‘2o) is thin. A special case of setcthin 48061. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (SetCat‘2o) ∈ ThinCat
Theorem	thincsect 48063	In a thin category, one morphism is a section of another iff they are pointing towards each other. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))
Theorem	thincsect2 48064	In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹))
Theorem	thincinv 48065	In a thin category, 𝐹 is an inverse of 𝐺 iff 𝐹 is a section of 𝐺 (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑆 = (Sect‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹(𝑋𝑆𝑌)𝐺))
Theorem	thinciso 48066	In a thin category, 𝐹:𝑋⟶𝑌 is an isomorphism iff there is a morphism from 𝑌 to 𝑋. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))
Theorem	thinccic 48067	In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅)))
21.46.13.2  Preordered sets as thin categories
Syntax	cprstc 48068	Class function defining preordered sets as categories.
class ProsetToCat
Definition	df-prstc 48069	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.)
⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Theorem	prstcval 48070	Lemma for prstcnidlem 48071 and prstcthin 48082. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Theorem	prstcnidlem 48071	Lemma for prstcnid 48072 and prstchomval 48080. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (comp‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
Theorem	prstcnid 48072	Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (comp‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx)    ⇒   ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶))
Theorem	prstcbas 48073	The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐾))    ⇒   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
Theorem	prstcleval 48074	Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    ⇒   ⊢ (𝜑 → ≤ = (le‘𝐶))
Theorem	prstclevalOLD 48075	Obsolete proof of prstcleval 48074 as of 12-Nov-2024. Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    ⇒   ⊢ (𝜑 → ≤ = (le‘𝐶))
Theorem	prstcle 48076	Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐾))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐶)𝑌))
Theorem	prstcocval 48077	Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ⊥ = (oc‘𝐾))    ⇒   ⊢ (𝜑 → ⊥ = (oc‘𝐶))
Theorem	prstcocvalOLD 48078	Obsolete proof of prstcocval 48077 as of 12-Nov-2024. Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ⊥ = (oc‘𝐾))    ⇒   ⊢ (𝜑 → ⊥ = (oc‘𝐶))
Theorem	prstcoc 48079	Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ⊥ = (oc‘𝐾))    ⇒   ⊢ (𝜑 → ( ⊥ ‘𝑋) = ((oc‘𝐶)‘𝑋))
Theorem	prstchomval 48080	Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    ⇒   ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))
Theorem	prstcprs 48081	The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 ∈ Proset )
Theorem	prstcthin 48082	The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	prstchom 48083	Hom-sets of the constructed category are dependent on the preorder. Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
Theorem	prstchom2 48084*	Hom-sets of the constructed category are dependent on the preorder. Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat ( see prstchom2ALT 48085). However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 21-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	prstchom2ALT 48085*	Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 48069. See prstchom2 48084 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → ≤ = (le‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
Theorem	postcpos 48086	The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
Theorem	postcposALT 48087	Alternate proof for postcpos 48086. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
Theorem	postc 48088*	The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦)))
21.46.13.3  Monoids as categories
Syntax	cmndtc 48089	Class function defining monoids as categories.
class MndToCat
Definition	df-mndtc 48090	Definition of the function converting a monoid to a category. Example 3.3(4.e) of [Adamek] p. 24. The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 48092), instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24. The resulting category is defined entirely, up to isomorphism, by mndtcbas 48093, mndtchom 48096, mndtcco 48097. Use those instead. See example 3.26(3) of [Adamek] p. 33 for more on isomorphism. "MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
Theorem	mndtcval 48091	Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
Theorem	mndtcbasval 48092	The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = {𝑀})
Theorem	mndtcbas 48093*	The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵)
Theorem	mndtcob 48094	Lemma for mndtchom 48096 and mndtcco 48097. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑀)
Theorem	mndtcbas2 48095	Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	mndtchom 48096	The only hom-set of the category built from a monoid is the base set of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (Base‘𝑀))
Theorem	mndtcco 48097	The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → · = (comp‘𝐶))    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (+g‘𝑀))
Theorem	mndtcco2 48098	The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → · = (comp‘𝐶))    &   ⊢ (𝜑 → ⚬ = (⟨𝑋, 𝑌⟩ · 𝑍))    ⇒   ⊢ (𝜑 → (𝐺 ⚬ 𝐹) = (𝐺(+g‘𝑀)𝐹))
Theorem	mndtccatid 48099*	Lemma for mndtccat 48100 and mndtcid 48101. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))
Theorem	mndtccat 48100	The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.)
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀))    &   ⊢ (𝜑 → 𝑀 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)