P. 502 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 50101-50200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opf11 50101	The object part of the op functor on functor categories. Lemma for fucoppc 50108. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (1st ‘(𝐹‘𝑋)) = (1st ‘𝑋))
Theorem	opf12 50102	The object part of the op functor on functor categories. Lemma for oppfdiag 50114. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝑀(2nd ‘(𝐹‘𝑋))𝑁) = (𝑁(2nd ‘𝑋)𝑀))
Theorem	opf2fval 50103*	The morphism part of the op functor on functor categories. Lemma for fucoppc 50108. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐹𝑌) = ( I ↾ (𝑌𝑁𝑋)))
Theorem	opf2 50104*	The morphism part of the op functor on functor categories. Lemma for fucoppc 50108. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (𝑌𝑁𝑋))    ⇒   ⊢ (𝜑 → ((𝑋𝐹𝑌)‘𝐶) = 𝐷)
Theorem	fucoppclem 50105	Lemma for fucoppc 50108. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝑌𝑁𝑋) = ((𝐹‘𝑋)(𝑂 Nat 𝑃)(𝐹‘𝑌)))
Theorem	fucoppcid 50106*	The opposite category of functors is compatible with the category of opposite functors in terms of identity morphism. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋)))
Theorem	fucoppcco 50107*	The opposite category of functors is compatible with the category of opposite functors in terms of composition. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ (𝑋(Hom ‘𝑅)𝑌))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑌(Hom ‘𝑅)𝑍))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝐵(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐴)) = (((𝑌𝐺𝑍)‘𝐵)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐴)))
Theorem	fucoppc 50108*	The isomorphism from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ 𝑇 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 𝐼 = (Iso‘𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹(𝑅𝐼𝑆)𝐺)
Theorem	fucoppcffth 50109*	A fully faithful functor from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐹((𝑅 Full 𝑆) ∩ (𝑅 Faith 𝑆))𝐺)
Theorem	fucoppcfunc 50110*	A functor from the opposite category of functors to the category of opposite functors. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑅 = (oppCat‘𝑄)    &   ⊢ 𝑆 = (𝑂 FuncCat 𝑃)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐶 Func 𝐷)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐶 Func 𝐷), 𝑦 ∈ (𝐶 Func 𝐷) ↦ ( I ↾ (𝑦𝑁𝑥))))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
Theorem	fucoppccic 50111	The opposite category of functors is isomorphic to the category of opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑋 = (oppCat‘(𝐷 FuncCat 𝐸))    &   ⊢ 𝑌 = ((oppCat‘𝐷) FuncCat (oppCat‘𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌)
Theorem	oppfdiag1 50112	A constant functor for opposite categories is the opposite functor of the constant functor for original categories. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))
Theorem	oppfdiag1a 50113	A constant functor for opposite categories is the opposite functor of the constant functor for original categories. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ( oppFunc ‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))
Theorem	oppfdiag 50114*	A diagonal functor for opposite categories is the opposite functor of the diagonal functor for original categories post-composed by an isomorphism (fucoppc 50108). (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ 𝑃 = (oppCat‘𝐷)    &   ⊢ 𝐿 = (𝐶Δfunc𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))    &   ⊢ 𝑁 = (𝐷 Nat 𝐶)    &   ⊢ (𝜑 → 𝐺 = (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛𝑁𝑚))))    ⇒   ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = (𝑂Δfunc𝑃))
21.50.16.2  Thin categories
Syntax	cthinc 50115	Extend class notation with the class of thin categories.
class ThinCat
Definition	df-thinc 50116*	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 50117*	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 50118*	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 50119*	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 50120	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
Theorem	thinccd 50121	A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	thincssc 50122	A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ThinCat ⊆ Cat
Theorem	isthincd2lem1 50123*	Lemma for isthincd2 50135 and thincmo2 50124. (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 50124	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	thinchom 50125	A non-empty hom-set of a thin category is given by its element. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = {𝐹})
Theorem	thincmo 50126*	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 50127*	Alternate proof of thincmo 50126. (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 50128*	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 50129*	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 50130	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 50131	In a thin category, all morphisms are monomorphisms. Example 7.33(9) of [Adamek] p. 110. The converse does not hold. See grptcmon 50291. (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 50132	In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 50292. (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 50133*	Lemma for isthincd2 50135. (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 50134*	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 50135*	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 50136	The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ 𝑂 = (oppCat‘𝐶)    ⇒   ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat)
Theorem	oppcthinco 50137	If the opposite category of a thin category has the same base and hom-sets as the original category, then it has the same composition operation as the original category. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂))    ⇒   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝑂))
Theorem	oppcthinendc 50138*	The opposite category of a thin category whose morphisms are all endomorphisms has the same base, hom-sets (oppcendc 49716) and composition operation as the original category. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≠ 𝑦 → (𝑥𝐻𝑦) = ∅))    ⇒   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝑂))
Theorem	oppcthinendcALT 50139*	Alternate proof of oppcthinendc 50138. (Contributed by Zhi Wang, 16-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑂 = (oppCat‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ ThinCat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≠ 𝑦 → (𝑥𝐻𝑦) = ∅))    ⇒   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝑂))
Theorem	thincpropd 50140	Two structures with the same base, hom-sets and composition operation are either both thin 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 ∈ ThinCat ↔ 𝐷 ∈ ThinCat))
Theorem	subthinc 50141	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 50142*	Lemma for functhinc 50146. 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 50143*	Lemma for functhinc 50146. (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 50144*	Lemma for functhinc 50146. 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 50145*	Lemma for functhinc 50146. 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 50146*	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 49710), and can be obtained from funcf2 17925, f002 49552, and ralrimivva 3214. (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	functhincfun 50147	A functor to a thin category is determined entirely by the object part. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ ThinCat)    ⇒   ⊢ (𝜑 → Fun (𝐶 Func 𝐷))
Theorem	fullthinc 50148*	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 50149	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 50150	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 50151*	Two thin categories are isomorphic iff the induced preorders are order-isomorphic. Example 3.26(2) of [Adamek] p. 33. Note that "thincciso.u" is redundant thanks to elbasfv 17275. (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	thinccisod 50152*	Two thin categories are isomorphic if the induced preorders are order-isomorphic (deduction form). Example 3.26(2) of [Adamek] p. 33. (Contributed by Zhi Wang, 22-Sep-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑋)    &   ⊢ 𝑆 = (Base‘𝑌)    &   ⊢ 𝐻 = (Hom ‘𝑋)    &   ⊢ 𝐽 = (Hom ‘𝑌)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ ThinCat)    &   ⊢ (𝜑 → 𝑌 ∈ ThinCat)    &   ⊢ (𝜑 → 𝐹:𝑅–1-1-onto→𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥𝐻𝑦) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))    ⇒   ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌)
Theorem	thincciso2 50153	Categories isomorphic to a thin category are thin. Example 3.26(2) of [Adamek] p. 33. Note that "thincciso2.u" is redundant thanks to elbasfv 17275. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    &   ⊢ (𝜑 → 𝑌 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝑋 ∈ ThinCat)
Theorem	thincciso3 50154	Categories isomorphic to a thin category are thin. Example 3.26(2) of [Adamek] p. 33. Note that "thincciso2.u" is redundant thanks to elbasfv 17275. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    &   ⊢ (𝜑 → 𝑋 ∈ ThinCat)    ⇒   ⊢ (𝜑 → 𝑌 ∈ ThinCat)
Theorem	thincciso4 50155	Two isomorphic categories are either both thin or neither. Note that "thincciso2.u" is redundant thanks to elbasfv 17275. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ 𝐶 = (CatCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌)    ⇒   ⊢ (𝜑 → (𝑋 ∈ ThinCat ↔ 𝑌 ∈ ThinCat))
Theorem	0thincg 50156	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 50157	The empty category (see 0cat 17745) is thin. (Contributed by Zhi Wang, 17-Sep-2024.)
⊢ ∅ ∈ ThinCat
Theorem	indcthing 50158*	An indiscrete category, i.e., a category where all hom-sets have exactly one morphism, is thin. (Contributed by Zhi Wang, 11-Nov-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = {𝐹})    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	discthing 50159*	A discrete category, i.e., a category where all morphisms are identity morphisms, is thin. Example 3.26(1) of [Adamek] p. 33. (Contributed by Zhi Wang, 11-Nov-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝐶))    &   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = if(𝑥 = 𝑦, {𝐼}, ∅))    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	indthinc 50160*	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 50162, 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 50161*	An alternate proof of indthinc 50160 assuming more axioms including ax-pow 5337 and ax-un 7733. (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 50162*	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 49709 and catprs2 49710 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 50163*	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 50164	The category (SetCat‘2o) is thin. A special case of setcthin 50163. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (SetCat‘2o) ∈ ThinCat
Theorem	thincsect 50165	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 50166	In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. Example 7.25(4) of [Adamek] p. 108. (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 50167	In a thin category, 𝐹 is an inverse of 𝐺 iff 𝐹 is a section of 𝐺. Example 7.20(7) of [Adamek] p. 107. (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 50168	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 50169	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) ≠ ∅)))
21.50.16.3  Terminal categories
Syntax	ctermc 50170	Extend class notation with the class of terminal categories.
class TermCat
Definition	df-termc 50171*	Definition of the proper class (termcnex 50274) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 50184). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24. As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50213). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50218). See also dftermc3 50229 where TermCat is defined as categories with exactly one disjointified arrow. Unlike https://ncatlab.org/nlab/show/terminal+category 50229, we reserve the term "trivial category" for (SetCat‘1o), justified by setc1oterm 50189. Followed directly from the definition, terminal categories are thin (termcthin 50175). The opposite category of a terminal category is "almost" itself (oppctermco 50203). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50238). Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50199) and terminal objects (dftermo4 50200). The universal properties provide an alternate proof of initoeu1 18068, termoeu1 18075, initoeu2 18073, and termoeu2 49936. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50214). The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17744, 0thincg 50156, func0g 49787, 0funcg 49783, and initc 49789. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
Theorem	istermc 50172*	The predicate "is a terminal category". A terminal category is a thin category with a singleton base set. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
Theorem	istermc2 50173*	The predicate "is a terminal category". A terminal category is a thin category with exactly one object. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃!𝑥 𝑥 ∈ 𝐵))
Theorem	istermc3 50174	The predicate "is a terminal category". A terminal category is a thin category whose base set is equinumerous to 1o. Consider en1b 9022, map1 9037, and euen1b 9025. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ 𝐵 ≈ 1o))
Theorem	termcthin 50175	A terminal category is a thin category. (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)
Theorem	termcthind 50176	A terminal category is a thin category (deduction form). (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ThinCat)
Theorem	termccd 50177	A terminal category is a category (deduction form). (Contributed by Zhi Wang, 16-Oct-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    ⇒   ⊢ (𝜑 → 𝐶 ∈ Cat)
Theorem	termcbas 50178*	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 50179	The object of a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)
⊢ (𝜑 → 𝐶 ∈ TermCat)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)
Theorem	termcbas2 50180	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 50181	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 50182	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 50183	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 50184	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 50185	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 50186	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 50187	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 50188	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 50189	The category (SetCat‘1o), i.e., the trivial category, is terminal. (Contributed by Zhi Wang, 18-Oct-2025.)
⊢ (SetCat‘1o) ∈ TermCat
Theorem	setc1obas 50190	The base of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    ⇒   ⊢ 1o = (Base‘ 1 )
Theorem	setc1ohomfval 50191	Set of morphisms of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    ⇒   ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘ 1 )
Theorem	setc1ocofval 50192	Composition in the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    ⇒   ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )
Theorem	setc1oid 50193	The identity morphism of the trivial category. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 1 = (SetCat‘1o)    &   ⊢ 𝐼 = (Id‘ 1 )    ⇒   ⊢ (𝐼‘∅) = ∅
Theorem	funcsetc1ocl 50194	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 50195*	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 50196	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 50197	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 50198	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 50199*	An alternate definition of df-inito 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.)
⊢ InitO = (𝑐 ∈ Cat ↦ ⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑐))‘∅) / 𝑓⦌dom (𝑓(𝑐 UP 𝑑)∅))
Theorem	dftermo4 50200*	An alternate definition of df-termo 18042 using universal property. See also the "Equivalent formulations" section of https://en.wikipedia.org/wiki/Initial_and_terminal_objects 18042. (Contributed by Zhi Wang, 23-Oct-2025.)
⊢ TermO = (𝑐 ∈ Cat ↦ ⦋(oppCat‘𝑐) / 𝑜⦌⦋(SetCat‘1o) / 𝑑⦌⦋((1st ‘(𝑑Δfunc𝑜))‘∅) / 𝑓⦌dom (𝑓(𝑜 UP 𝑑)∅))