P. 503 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50201-50300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	termcthind 50201	A terminal category is a thin category (deduction form). (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	termccd 50202	A terminal category is a category (deduction form). (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	termcbas 50203*	The base of a terminal category is a singleton. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
Theorem	termco 50204	The object of a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)
Theorem	termcbas2 50205	The base of a terminal category is given by its object. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 = {𝑋})
Theorem	termcbasmo 50206	Two objects in a terminal category are identical. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	termchomn0 50207	All hom-sets of a terminal category are non-empty. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)
Theorem	termchommo 50208	All morphisms of a terminal category are identical. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑊))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	termcid 50209	The morphism of a terminal category is an identity morphism. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋))
Theorem	termcid2 50210	The morphism of a terminal category is an identity morphism. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑌))
Theorem	termchom 50211	The hom-set of a terminal category is a singleton of the identity morphism. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})
Theorem	termchom2 50212	The hom-set of a terminal category is a singleton of the identity morphism. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑍)})
Theorem	setcsnterm 50213	The category of one set, either a singleton set or an empty set, is terminal. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ (SetCat‘{{𝐴}}) ∈ TermCat
Theorem	setc1oterm 50214	The category (SetCat‘1o), i.e., the trivial category, is terminal. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ (SetCat‘1o) ∈ TermCat
Theorem	setc1obas 50215	The base of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    ⇒   ⊢ 1o = (Base‘ 1 )
Theorem	setc1ohomfval 50216	Set of morphisms of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    ⇒   ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘ 1 )
Theorem	setc1ocofval 50217	Composition in the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    ⇒   ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )
Theorem	setc1oid 50218	The identity morphism of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    &   ⊢ 𝐼 = (Id‘ 1 )    ⇒   ⊢ (𝐼‘∅) = ∅
Theorem	funcsetc1ocl 50219	The functor to the trivial category. The converse is also true due to reverse closure. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    &   ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))
Theorem	funcsetc1o 50220*	Value of the functor to the trivial category. The converse is also true because 𝐹 would be the empty set if 𝐶 were not a category; and the empty set cannot equal an ordered pair of two sets. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    &   ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐹 = ⟨(𝐵 × 1o), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × 1o))⟩)
Theorem	isinito2lem 50221	The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ 1 = (SetCat‘1o)    &   ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐼 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(𝐹(𝐶 UP 1 )∅)∅))
Theorem	isinito2 50222	The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ 1 = (SetCat‘1o)    &   ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    ⇒   ⊢ (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(𝐹(𝐶 UP 1 )∅)∅)
Theorem	isinito3 50223	The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ 1 = (SetCat‘1o)    &   ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)    ⇒   ⊢ (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ dom (𝐹(𝐶 UP 1 )∅))
Theorem	dfinito4 50224*	An alternate definition of df-inito 18040 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 18040. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))
Theorem	dftermo4 50225*	An alternate definition of df-termo 18041 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 18041. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ TermO = (𝑐 ∈ Cat ↦ ⦋(oppCat‘𝑐) / 𝑜⦌⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑜))‘∅) / 𝑓⦌dom (𝑓(𝑜 UP 𝑑)∅))
Theorem	termcpropd 50226	Two structures with the same base, hom-sets and composition operation are either both terminal categories or neither. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐶 ∈ TermCat ↔ 𝐷 ∈ TermCat))
Theorem	oppctermhom 50227	The opposite category of a terminal category has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ TermCat)    ⇒   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂))
Theorem	oppctermco 50228	The opposite category of a terminal category has the same base, hom-sets and composition operation as the original category. Note that 𝐶 = 𝑂 cannot be proved because 𝐶 might not even be a function. For example, let 𝐶 be ({⟨(Base‘ndx), {∅}⟩, ⟨(Hom ‘ndx), ((V × V) × {{∅}})⟩} ∪ {⟨(comp‘ndx), {∅}⟩, ⟨(comp‘ndx), 2o⟩}); it should be a terminal category, but the opposite category is not itself. See the definitions df-oppc 17767 and df-sets 17223. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ TermCat)    ⇒   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝑂))
Theorem	oppcterm 50229	The opposite category of a terminal category is a terminal category. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ TermCat)    ⇒   ⊢ (𝜑 → 𝑂 ∈ TermCat)
Theorem	functermclem 50230	Lemma for functermc 50231. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ ((𝜑 ∧ 𝐾𝑅𝐿) → 𝐾 = 𝐹)    &   ⊢ (𝜑 → (𝐹𝑅𝐿 ↔ 𝐿 = 𝐺))    ⇒   ⊢ (𝜑 → (𝐾𝑅𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))
Theorem	functermc 50231*	Functor to a terminal category. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 𝐹 = (𝐵 × 𝐶)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    ⇒   ⊢ (𝜑 → (𝐾(𝐷 Func 𝐸)𝐿 ↔ (𝐾 = 𝐹 ∧ 𝐿 = 𝐺)))
Theorem	functermc2 50232*	Functor to a terminal category. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ 𝐶 = (Base‘𝐸)    &   ⊢ 𝐻 = (Hom ‘𝐷)    &   ⊢ 𝐽 = (Hom ‘𝐸)    &   ⊢ 𝐹 = (𝐵 × 𝐶)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    ⇒   ⊢ (𝜑 → (𝐷 Func 𝐸) = {⟨𝐹, 𝐺⟩})
Theorem	functermceu 50233*	There exists a unique functor to a terminal category. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷))
Theorem	fulltermc 50234*	A functor to a terminal category is full iff all hom-sets of the source category are non-empty. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅))
Theorem	fulltermc2 50235	Given a full functor to a terminal category, the source category must not have empty hom-sets. (Contributed by Zhi Wang, 17-Oct-2025.) (Proof shortened by Zhi Wang, 6-Nov-2025.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)
Theorem	termcterm 50236	A terminal category is a terminal object of the category of small categories. (Contributed by Zhi Wang, 17-Oct-2025.)
⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ TermCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ (TermO‘𝐸))
Theorem	termcterm2 50237	A terminal object of the category of small categories is a terminal category. (Contributed by Zhi Wang, 18-Oct-2025.) (Proof shortened by Zhi Wang, 23-Oct-2025.)
⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → (𝑈 ∩ TermCat) ≠ ∅)    &   ⊢ (𝜑 → 𝐶 ∈ (TermO‘𝐸))    ⇒   ⊢ (𝜑 → 𝐶 ∈ TermCat)
Theorem	termcterm3 50238	In the category of small categories, a terminal object is equivalent to a terminal category. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → (SetCat‘1o) ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐶 ∈ TermCat ↔ 𝐶 ∈ (TermO‘𝐸)))
Theorem	termcciso 50239	A category is isomorphic to a terminal category iff it itself is terminal. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ TermCat)    ⇒   ⊢ (𝜑 → (𝑌 ∈ TermCat ↔ 𝑋( ≃𝑐 ‘𝐶)𝑌))
Theorem	termccisoeu 50240*	The isomorphism between terminal categories is unique. (Contributed by Zhi Wang, 26-Oct-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ TermCat)    &   ⊢ (𝜑 → 𝑌 ∈ TermCat)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝑋(Iso‘𝐶)𝑌))
Theorem	termc2 50241*	If there exists a unique functor from both the category itself and the trivial category, then the category is terminal. Note that the converse also holds, so that it is a biconditional. See the proof of termc 50242 for hints. See also eufunc 50245 and euendfunc2 50250 for some insights on why two categories are sufficient. (Contributed by Zhi Wang, 18-Oct-2025.) (Proof shortened by Zhi Wang, 20-Oct-2025.)
⊢ (∀𝑑 ∈ ({𝐶, (SetCat‘1o)} ∩ Cat)∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶) → 𝐶 ∈ TermCat)
Theorem	termc 50242*	Alternate definition of TermCat. See also df-termc 50196. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ (𝐶 ∈ TermCat ↔ ∀𝑑 ∈ Cat ∃!𝑓 𝑓 ∈ (𝑑 Func 𝐶))
Theorem	dftermc2 50243*	Alternate definition of TermCat. See also df-termc 50196 and dftermc3 50254. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ TermCat = {𝑐 ∣ ∀𝑑 ∈ Cat ∃!𝑓 𝑓 ∈ (𝑑 Func 𝑐)}
Theorem	eufunclem 50244*	If there exists a unique functor from a non-empty category, then the base of the target category is at most a singleton. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ 𝐵 = (Base‘𝐷)    ⇒   ⊢ (𝜑 → 𝐵 ≼ 1o)
Theorem	eufunc 50245*	If there exists a unique functor from a non-empty category, then the base of the target category is a singleton. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐷))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ 𝐵 = (Base‘𝐷)    ⇒   ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵)
Theorem	idfudiag1lem 50246	Lemma for idfudiag1bas 50247 and idfudiag1 50248. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ (𝜑 → ( I ↾ 𝐴) = (𝐴 × {𝐵}))    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → 𝐴 = {𝐵})
Theorem	idfudiag1bas 50247	If the identity functor of a category is the same as a constant functor to the category, then the base is a singleton. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐿 = (𝐶Δfunc𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ (𝜑 → 𝐼 = 𝐾)    ⇒   ⊢ (𝜑 → 𝐵 = {𝑋})
Theorem	idfudiag1 50248	If the identity functor of a category is the same as a constant functor to the category, then the category is terminal. (Contributed by Zhi Wang, 19-Oct-2025.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐿 = (𝐶Δfunc𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)    &   ⊢ (𝜑 → 𝐼 = 𝐾)    ⇒   ⊢ (𝜑 → 𝐶 ∈ TermCat)
Theorem	euendfunc 50249*	If there exists a unique endofunctor (a functor from a category to itself) for a non-empty category, then the category is terminal. This partially explains why two categories are sufficient in termc2 50241. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐶 Func 𝐶))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    ⇒   ⊢ (𝜑 → 𝐶 ∈ TermCat)
Theorem	euendfunc2 50250	If there exists a unique endofunctor (a functor from a category to itself) for a category, then it is either initial (empty) or terminal. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ ((𝐶 Func 𝐶) ≈ 1o → ((Base‘𝐶) = ∅ ∨ 𝐶 ∈ TermCat))
Theorem	termcarweu 50251*	There exists a unique disjointified arrow in a terminal category. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝐶 ∈ TermCat → ∃!𝑎 𝑎 ∈ (Arrow‘𝐶))
Theorem	arweuthinc 50252*	If a structure has a unique disjointified arrow, then the structure is a thin category. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ ThinCat)
Theorem	arweutermc 50253*	If a structure has a unique disjointified arrow, then the structure is a terminal category. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (∃!𝑎 𝑎 ∈ (Arrow‘𝐶) → 𝐶 ∈ TermCat)
Theorem	dftermc3 50254	Alternate definition of TermCat. See also df-termc 50196, dftermc2 50243. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ TermCat = {𝑐 ∣ (Arrow‘𝑐) ≈ 1o}
Theorem	termcfuncval 50255	The value of a functor from a terminal category. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑋 = ((1st ‘𝐾)‘𝑌)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝐼 = (Id‘𝐷)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ 𝐾 = ⟨{⟨𝑌, 𝑋⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩}⟩))
Theorem	diag1f1olem 50256	To any functor from a terminal category can an object in the target base be assigned. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑋 = ((1st ‘𝐾)‘𝑌)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ 𝐾 = ((1st ‘𝐿)‘𝑋)))
Theorem	diag1f1o 50257	The object part of the diagonal functor is a bijection if 𝐷 is terminal. So any functor from a terminal category is one-to-one correspondent to an object of the target base. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    ⇒   ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1-onto→(𝐷 Func 𝐶))
Theorem	termcnatval 50258	Value of natural transformations for a terminal category. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑅 = (𝐴‘𝑋)    ⇒   ⊢ (𝜑 → 𝐴 = {⟨𝑋, 𝑅⟩})
Theorem	diag2f1olem 50259	Lemma for diag2f1o 50260. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝑀 ∈ (((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))    &   ⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ 𝐹 = (𝑀‘𝑍)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹)))
Theorem	diag2f1o 50260	If 𝐷 is terminal, the morphism part of a diagonal functor is bijective functions from hom-sets into sets of natural transformations. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1-onto→(((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))
Theorem	diagffth 50261	The diagonal functor is a fully faithful functor from a category 𝐶 to the category of functors from a terminal category to 𝐶. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐶)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    ⇒   ⊢ (𝜑 → 𝐿 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
Theorem	diagciso 50262	The diagonal functor is an isomorphism from a category 𝐶 to the category of functors from a terminal category to 𝐶. It is provable that the inverse of the diagonal functor is the mapped object by the transposed curry of (𝐷 evalF 𝐶), i.e., ∪ ran (1st ‘(⟨𝐷, 𝑄⟩ curryF ((𝐷 evalF 𝐶) ∘func (𝐷 swapF 𝑄)))). (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐶)    &   ⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑈)    &   ⊢ 𝐼 = (Iso‘𝐸)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (𝐶𝐼𝑄))
Theorem	diagcic 50263	Any category 𝐶 is isomorphic to the category of functors from a terminal category to 𝐶. See also the "Properties" section of https://ncatlab.org/nlab/show/terminal+category. Therefore the number of categories isomorphic to a non-empty category is at least the number of singletons, so large (snnex 7756) that these isomorphic categories form a proper class. (Contributed by Zhi Wang, 21-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ 𝑄 = (𝐷 FuncCat 𝐶)    &   ⊢ 𝐸 = (CatCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝐶( ≃𝑐 ‘𝐸)𝑄)
Theorem	funcsn 50264	The category of one functor to a thin category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐶 Func 𝐷) = {𝐹})    &   ⊢ (𝜑 → 𝐷 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝑄 ∈ TermCat)
Theorem	fucterm 50265	The category of functors to a terminal category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    ⇒   ⊢ (𝜑 → 𝑄 ∈ TermCat)
Theorem	0fucterm 50266	The category of functors from an initial category is terminal. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ∅ = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    ⇒   ⊢ (𝜑 → 𝑄 ∈ TermCat)
Theorem	termfucterm 50267	All functors between two terminal categories are isomorphisms. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ TermCat)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ TermCat)    ⇒   ⊢ (𝜑 → (𝑋 Func 𝑌) = (𝑋𝐼𝑌))
Theorem	cofuterm 50268	Post-compose with a functor to a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → 𝐸 ∈ TermCat)    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐾)
Theorem	uobeqterm 50269	Universal objects and terminal categories. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ 𝐴 = (Base‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ TermCat)    &   ⊢ (𝜑 → 𝐸 ∈ TermCat)    ⇒   ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))
Theorem	isinito4 50270	The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 1 ∈ TermCat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘ 1 ))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))    ⇒   ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ dom (𝐹(𝐶 UP 1 )𝑋)))
Theorem	isinito4a 50271	The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 1 ∈ TermCat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘ 1 ))    &   ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘𝑋)    ⇒   ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ dom (𝐹(𝐶 UP 1 )𝑋)))
21.50.16.4  Preordered sets as thin categories
Syntax	cprstc 50272	Class function defining preordered sets as categories.
class ProsetToCat
Definition	df-prstc 50273	Definition of the function converting a preordered set to a category. Justified by prsthinc 50187. 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 50176), by prstcnid 50276, prstchom 50285, and prstcthin 50284. Other important properties include prstcbas 50277, prstcleval 50278, prstcle 50279, prstcocval 50280, prstcoc 50281, prstchom2 50286, and prstcprs 50283. Use those instead. Note that the defining property prstchom 50285 is equivalent to prstchom2 50286 given prstcthin 50284. See thincn0eu 50154 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 50274	Lemma for prstcnidlem 50275 and prstcthin 50284. (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 50275	Lemma for prstcnid 50276 and prstchomval 50282. (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 50276	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 50277	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 50278	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	prstcle 50279	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 50280	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	prstcoc 50281	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 50282	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 50283	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 50284	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 50285	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 50286*	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 50287). 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 50287*	Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 50273. See prstchom2 50286 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	oduoppcbas 50288	The dual of a preordered set and the opposite category have the same set of objects. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → 𝐷 = (ProsetToCat‘(ODual‘𝐾)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝜑 → (Base‘𝐷) = (Base‘𝑂))
Theorem	oduoppcciso 50289	The dual of a preordered set and the opposite category are category-isomorphic. Example 3.6(1) of [Adamek] p. 25. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Proset )    &   ⊢ (𝜑 → 𝐷 = (ProsetToCat‘(ODual‘𝐾)))    &   ⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝐷( ≃𝑐 ‘(CatCat‘𝑈))𝑂)
Theorem	postcpos 50290	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 50291	Alternate proof of postcpos 50290. (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 50292*	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)))
Theorem	discsntermlem 50293*	A singlegon is an element of the class of singlegons. The converse (basrestermcfolem 50294) also holds. This is trivial if 𝐵 is 𝑏 (abid 2743). (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (∃𝑥 𝐵 = {𝑥} → 𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}})
Theorem	basrestermcfolem 50294*	An element of the class of singlegons is a singlegon. The converse (discsntermlem 50293) also holds. This is trivial if 𝐵 is 𝑏 (abid 2743). (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝐵 ∈ {𝑏 ∣ ∃𝑥 𝑏 = {𝑥}} → ∃𝑥 𝐵 = {𝑥})
Theorem	discbas 50295	A discrete category (a category whose only morphisms are the identity morphisms) can be constructed for any base set. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   ⊢ 𝐶 = (ProsetToCat‘𝐾)    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐶))
Theorem	discthin 50296	A discrete category (a category whose only morphisms are the identity morphisms) is thin. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   ⊢ 𝐶 = (ProsetToCat‘𝐾)    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐶 ∈ ThinCat)
Theorem	discsnterm 50297*	A discrete category (a category whose only morphisms are the identity morphisms) with a singlegon base is terminal. Corollary of example 3.3(4)(c) of [Adamek] p. 24 and example 3.26(1) of [Adamek] p. 33. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ( I ↾ 𝐵)⟩}    &   ⊢ 𝐶 = (ProsetToCat‘𝐾)    ⇒   ⊢ (∃𝑥 𝐵 = {𝑥} → 𝐶 ∈ TermCat)
Theorem	basrestermcfo 50298*	The base function restricted to the class of terminal categories maps the class of terminal categories onto the class of singletons. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (Base ↾ TermCat):TermCat–onto→{𝑏 ∣ ∃𝑥 𝑏 = {𝑥}}
Theorem	termcnex 50299	The class of all terminal categories is a proper class. Therefore both the class of all thin categories and the class of all categories are proper classes. Note that snnex 7756 is equivalent to sngl V ∉ V. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ TermCat ∉ V
21.50.16.5  Monoids as categories
Syntax	cmndtc 50300	Class function defining monoids as categories.
class MndToCat