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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisepi2 17001* Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝐸 = (Epi‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑌𝐻𝑧)∀ ∈ (𝑌𝐻𝑧)((𝑔(⟨𝑋, 𝑌· 𝑧)𝐹) = ((⟨𝑋, 𝑌· 𝑧)𝐹) → 𝑔 = ))))
 
Theoremepihom 17002 An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝐸 = (Epi‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋𝐻𝑌))
 
Theoremepii 17003 Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝐸 = (Epi‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐸𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &   (𝜑𝐾 ∈ (𝑌𝐻𝑍))       (𝜑 → ((𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐾(⟨𝑋, 𝑌· 𝑍)𝐹) ↔ 𝐺 = 𝐾))
 
8.1.4  Sections, inverses, isomorphisms
 
Syntaxcsect 17004 Extend class notation with the sections of a morphism.
class Sect
 
Syntaxcinv 17005 Extend class notation with the inverses of a morphism.
class Inv
 
Syntaxciso 17006 Extend class notation with the class of all isomorphisms.
class Iso
 
Definitiondf-sect 17007* Function returning the section relation in a category. Given arrows 𝑓:𝑋𝑌 and 𝑔:𝑌𝑋, we say 𝑓Sect𝑔, that is, 𝑓 is a section of 𝑔, if 𝑔𝑓 = 1‘𝑋. If there there is an arrow 𝑔 with 𝑓Sect𝑔, the arrow 𝑓 is called a section, see definition 7.19 of [Adamek] p. 106. (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ]((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}))
 
Definitiondf-inv 17008* The inverse relation in a category. Given arrows 𝑓:𝑋𝑌 and 𝑔:𝑌𝑋, we say 𝑔Inv𝑓, that is, 𝑔 is an inverse of 𝑓, if 𝑔 is a section of 𝑓 and 𝑓 is a section of 𝑔. Definition 3.8 of [Adamek] p. 28. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
 
Definitiondf-iso 17009* Function returning the isomorphisms of the category 𝑐. Definition 3.8 of [Adamek] p. 28, and definition in [Lang] p. 54. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 2-Jan-2017.)
Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
 
Theoremsectffval 17010* Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    1 = (Id‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑𝑆 = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
 
Theoremsectfval 17011* Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    1 = (Id‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))})
 
Theoremsectss 17012 The section relation is a relation between morphisms from 𝑋 to 𝑌 and morphisms from 𝑌 to 𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    1 = (Id‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
 
Theoremissect 17013 The property "𝐹 is a section of 𝐺". (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    1 = (Id‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋))))
 
Theoremissect2 17014 Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    1 = (Id‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑋))       (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)))
 
Theoremsectcan 17015 If 𝐺 is a section of 𝐹 and 𝐹 is a section of 𝐻, then 𝐺 = 𝐻. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐺(𝑋𝑆𝑌)𝐹)    &   (𝜑𝐹(𝑌𝑆𝑋)𝐻)       (𝜑𝐺 = 𝐻)
 
Theoremsectco 17016 Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &    · = (comp‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹(𝑋𝑆𝑌)𝐺)    &   (𝜑𝐻(𝑌𝑆𝑍)𝐾)       (𝜑 → (𝐻(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(⟨𝑍, 𝑌· 𝑋)𝐾))
 
Theoremisofval 17017* Function value of the function returning the isomorphisms of a category. (Contributed by AV, 5-Apr-2017.)
(𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
 
Theoreminvffval 17018* Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑆 = (Sect‘𝐶)       (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥𝑆𝑦) ∩ (𝑦𝑆𝑥))))
 
Theoreminvfval 17019 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑆 = (Sect‘𝐶)       (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ (𝑌𝑆𝑋)))
 
Theoremisinv 17020 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑆 = (Sect‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺𝐺(𝑌𝑆𝑋)𝐹)))
 
Theoreminvss 17021 The inverse relation is a relation between morphisms 𝐹:𝑋𝑌 and their inverses 𝐺:𝑌𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)))
 
Theoreminvsym 17022 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺𝐺(𝑌𝑁𝑋)𝐹))
 
Theoreminvsym2 17023 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑(𝑋𝑁𝑌) = (𝑌𝑁𝑋))
 
Theoreminvfun 17024 The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → Fun (𝑋𝑁𝑌))
 
Theoremisoval 17025 The isomorphisms are the domain of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 21-May-2020.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
 
Theoreminviso1 17026 If 𝐺 is an inverse to 𝐹, then 𝐹 is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)    &   (𝜑𝐹(𝑋𝑁𝑌)𝐺)       (𝜑𝐹 ∈ (𝑋𝐼𝑌))
 
Theoreminviso2 17027 If 𝐺 is an inverse to 𝐹, then 𝐺 is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)    &   (𝜑𝐹(𝑋𝑁𝑌)𝐺)       (𝜑𝐺 ∈ (𝑌𝐼𝑋))
 
Theoreminvf 17028 The inverse relation is a function from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
 
Theoreminvf1o 17029 The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism 𝐹 ∈ (𝑋𝐼𝑌) has a unique inverse, denoted by ((Inv‘𝐶)‘𝐹). Remark 3.12 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋))
 
Theoreminvinv 17030 The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))       (𝜑 → ((𝑌𝑁𝑋)‘((𝑋𝑁𝑌)‘𝐹)) = 𝐹)
 
Theoreminvco 17031 The composition of two isomorphisms is an isomorphism, and the inverse is the composition of the individual inverses. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))    &    · = (comp‘𝐶)    &   (𝜑𝑍𝐵)    &   (𝜑𝐺 ∈ (𝑌𝐼𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹)(𝑋𝑁𝑍)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑍, 𝑌· 𝑋)((𝑌𝑁𝑍)‘𝐺)))
 
Theoremdfiso2 17032* Alternate definition of an isomorphism of a category, according to definition 3.8 in [Adamek] p. 28. (Contributed by AV, 10-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐼 = (Iso‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &    1 = (Id‘𝐶)    &    = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)    &    = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 𝐹) = ( 1𝑋) ∧ (𝐹 𝑔) = ( 1𝑌))))
 
Theoremdfiso3 17033* Alternate definition of an isomorphism of a category as a section in both directions. (Contributed by AV, 11-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹𝐹(𝑋𝑆𝑌)𝑔)))
 
Theoreminveq 17034 If there are two inverses of a morphism, these inverses are equal. Corollary 3.11 of [Adamek] p. 28. (Contributed by AV, 10-Apr-2020.) (Revised by AV, 3-Jul-2022.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))
 
Theoremisofn 17035 The function value of the function returning the isomorphisms of a category is a function over the square product of the base set of the category. (Contributed by AV, 5-Apr-2020.)
(𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
 
Theoremisohom 17036 An isomorphism is a homomorphism. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋𝐻𝑌))
 
Theoremisoco 17037 The composition of two isomorphisms is an isomorphism. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &    · = (comp‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐼𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐼𝑍))
 
Theoremoppcsect 17038 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑆 = (Sect‘𝐶)    &   𝑇 = (Sect‘𝑂)       (𝜑 → (𝐹(𝑋𝑇𝑌)𝐺𝐺(𝑋𝑆𝑌)𝐹))
 
Theoremoppcsect2 17039 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑆 = (Sect‘𝐶)    &   𝑇 = (Sect‘𝑂)       (𝜑 → (𝑋𝑇𝑌) = (𝑋𝑆𝑌))
 
Theoremoppcinv 17040 An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Inv‘𝐶)    &   𝐽 = (Inv‘𝑂)       (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))
 
Theoremoppciso 17041 An isomorphism in the opposite category. See also remark 3.9 in [Adamek] p. 28. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)    &   𝐽 = (Iso‘𝑂)       (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))
 
Theoremsectmon 17042 If 𝐹 is a section of 𝐺, then 𝐹 is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑀 = (Mono‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹(𝑋𝑆𝑌)𝐺)       (𝜑𝐹 ∈ (𝑋𝑀𝑌))
 
Theoremmonsect 17043 If 𝐹 is a monomorphism and 𝐺 is a section of 𝐹, then 𝐺 is an inverse of 𝐹 and they are both isomorphisms. This is also stated as "a monomorphism which is also a split epimorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝑀 = (Mono‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐹 ∈ (𝑋𝑀𝑌))    &   (𝜑𝐺(𝑌𝑆𝑋)𝐹)       (𝜑𝐹(𝑋𝑁𝑌)𝐺)
 
Theoremsectepi 17044 If 𝐹 is a section of 𝐺, then 𝐺 is an epimorphism. An epimorphism that arises from a section is also known as a split epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐸 = (Epi‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹(𝑋𝑆𝑌)𝐺)       (𝜑𝐺 ∈ (𝑌𝐸𝑋))
 
Theoremepisect 17045 If 𝐹 is an epimorphism and 𝐹 is a section of 𝐺, then 𝐺 is an inverse of 𝐹 and they are both isomorphisms. This is also stated as "an epimorphism which is also a split monomorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐸 = (Epi‘𝐶)    &   𝑆 = (Sect‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐸𝑌))    &   (𝜑𝐹(𝑋𝑆𝑌)𝐺)       (𝜑𝐹(𝑋𝑁𝑌)𝐺)
 
Theoremsectid 17046 The identity is a section of itself. (Contributed by AV, 8-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋)(𝑋(Sect‘𝐶)𝑋)(𝐼𝑋))
 
Theoreminvid 17047 The inverse of the identity is the identity. (Contributed by AV, 8-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋)(𝑋(Inv‘𝐶)𝑋)(𝐼𝑋))
 
Theoremidiso 17048 The identity is an isomorphism. Example 3.13 of [Adamek] p. 28. (Contributed by AV, 8-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋) ∈ (𝑋(Iso‘𝐶)𝑋))
 
Theoremidinv 17049 The inverse of the identity is the identity. Example 3.13 of [Adamek] p. 28. (Contributed by AV, 9-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ((𝑋(Inv‘𝐶)𝑋)‘(𝐼𝑋)) = (𝐼𝑋))
 
Theoreminvisoinvl 17050 The inverse of an isomorphism 𝐹 (which is unique because of invf 17028 and is therefore denoted by ((𝑋𝑁𝑌)‘𝐹), see also remark 3.12 in [Adamek] p. 28) is invers to the isomorphism. (Contributed by AV, 9-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))       (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
 
Theoreminvisoinvr 17051 The inverse of an isomorphism is invers to the isomorphism. (Contributed by AV, 9-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))       (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
 
Theoreminvcoisoid 17052 The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 5-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))    &    1 = (Id‘𝐶)    &    = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)       (𝜑 → (((𝑋𝑁𝑌)‘𝐹) 𝐹) = ( 1𝑋))
 
Theoremisocoinvid 17053 The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))    &    1 = (Id‘𝐶)    &    = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)       (𝜑 → (𝐹 ((𝑋𝑁𝑌)‘𝐹)) = ( 1𝑌))
 
Theoremrcaninv 17054 Right cancellation of an inverse of an isomorphism. (Contributed by AV, 5-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))    &   (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))    &   (𝜑𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))    &   𝑅 = ((𝑌𝑁𝑋)‘𝐹)    &    = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)       (𝜑 → ((𝐺 𝑅) = (𝐻 𝑅) → 𝐺 = 𝐻))
 
8.1.5  Isomorphic objects

In this subsection, the "is isomorphic to" relation between objects of a category 𝑐 is defined (see df-cic 17056). It is shown that this relation is an equivalence relation, see cicer 17066.

 
Syntaxccic 17055 Extend class notation to include the category isomorphism relation.
class 𝑐
 
Definitiondf-cic 17056 Function returning the set of isomorphic objects for each category 𝑐. Definition 3.15 of [Adamek] p. 29. Analogous to the definition of the group isomorphism relation 𝑔, see df-gic 18340. (Contributed by AV, 4-Apr-2020.)
𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
 
Theoremcicfval 17057 The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.)
(𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
 
Theorembrcic 17058 The relation "is isomorphic to" for categories. (Contributed by AV, 5-Apr-2020.)
𝐼 = (Iso‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅))
 
Theoremcic 17059* Objects 𝑋 and 𝑌 in a category are isomorphic provided that there is an isomorphism 𝑓:𝑋𝑌, see definition 3.15 of [Adamek] p. 29. (Contributed by AV, 4-Apr-2020.)
𝐼 = (Iso‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌)))
 
Theorembrcici 17060 Prove that two objects are isomorphic by an explicit isomorphism. (Contributed by AV, 4-Apr-2020.)
𝐼 = (Iso‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))       (𝜑𝑋( ≃𝑐𝐶)𝑌)
 
Theoremcicref 17061 Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐𝐶)𝑂)
 
Theoremciclcl 17062 Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
 
Theoremcicrcl 17063 Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
 
Theoremcicsym 17064 Isomorphism is symmetric. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆( ≃𝑐𝐶)𝑅)
 
Theoremcictr 17065 Isomorphism is transitive. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆𝑆( ≃𝑐𝐶)𝑇) → 𝑅( ≃𝑐𝐶)𝑇)
 
Theoremcicer 17066 Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.)
(𝐶 ∈ Cat → ( ≃𝑐𝐶) Er (Base‘𝐶))
 
8.1.6  Subcategories
 
Syntaxcssc 17067 Extend class notation to include the subset relation for subcategories.
class cat
 
Syntaxcresc 17068 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
class cat
 
Syntaxcsubc 17069 Extend class notation to include the collection of subcategories of a category.
class Subcat
 
Definitiondf-ssc 17070* Define the subset relation for subcategories. Despite the name, this is not really a "category-aware" definition, which is to say it makes no explicit references to homsets or composition; instead this is a subset-like relation on the functions that are used as subcategory specifications in df-subc 17072, which makes it play an analogous role to the subset relation applied to the subgroups of a group. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat = {⟨, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗𝑥))}
 
Definitiondf-resc 17071* Define the restriction of a category to a given set of arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat = (𝑐 ∈ V, ∈ V ↦ ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩))
 
Definitiondf-subc 17072* (Subcat‘𝐶) is the set of all the subcategory specifications of the category 𝐶. Like df-subg 18216, this is not actually a collection of categories (as in definition 4.1(a) of [Adamek] p. 48), but only sets which when given operations from the base category (using df-resc 17071) form a category. All the objects and all the morphisms of the subcategory belong to the supercategory. The identity of an object, the domain and the codomain of a morphism are the same in the subcategory and the supercategory. The composition of the subcategory is a restriction of the composition of the supercategory. (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 4-Jan-2017.)
Subcat = (𝑐 ∈ Cat ↦ { ∣ (cat (Homf𝑐) ∧ [dom dom / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑧)))})
 
Theoremsscrel 17073 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Rel ⊆cat
 
Theorembrssc 17074* The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
 
Theoremsscpwex 17075* An analogue of pwex 5273 for the subcategory subset relation: The collection of subcategory subsets of a given set 𝐽 is a set. (Contributed by Mario Carneiro, 6-Jan-2017.)
{cat 𝐽} ∈ V
 
Theoremsubcrcl 17076 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
 
Theoremsscfn1 17077 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻cat 𝐽)    &   (𝜑𝑆 = dom dom 𝐻)       (𝜑𝐻 Fn (𝑆 × 𝑆))
 
Theoremsscfn2 17078 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻cat 𝐽)    &   (𝜑𝑇 = dom dom 𝐽)       (𝜑𝐽 Fn (𝑇 × 𝑇))
 
Theoremssclem 17079 Lemma for ssc1 17081 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻 Fn (𝑆 × 𝑆))       (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V))
 
Theoremisssc 17080* Value of the subcategory subset relation when the arguments are known functions. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝐽 Fn (𝑇 × 𝑇))    &   (𝜑𝑇𝑉)       (𝜑 → (𝐻cat 𝐽 ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
 
Theoremssc1 17081 Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝐽 Fn (𝑇 × 𝑇))    &   (𝜑𝐻cat 𝐽)       (𝜑𝑆𝑇)
 
Theoremssc2 17082 Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝐻cat 𝐽)    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
 
Theoremsscres 17083 Any function restricted to a square domain is a subcategory subset of the original. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)
 
Theoremsscid 17084 The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝐻cat 𝐻)
 
Theoremssctr 17085 The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)
 
Theoremssceq 17086 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)
 
Theoremrescval 17087 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)       ((𝐶𝑉𝐻𝑊) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
 
Theoremrescval2 17088 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   (𝜑𝐶𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))       (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
 
Theoremrescbas 17089 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑆𝐵)       (𝜑𝑆 = (Base‘𝐷))
 
Theoremreschom 17090 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑆𝐵)       (𝜑𝐻 = (Hom ‘𝐷))
 
Theoremreschomf 17091 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑆𝐵)       (𝜑𝐻 = (Homf𝐷))
 
Theoremrescco 17092 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝑆𝐵)    &    · = (comp‘𝐶)       (𝜑· = (comp‘𝐷))
 
Theoremrescabs 17093 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐶𝑉)    &   (𝜑𝐻 Fn (𝑆 × 𝑆))    &   (𝜑𝐽 Fn (𝑇 × 𝑇))    &   (𝜑𝑆𝑊)    &   (𝜑𝑇𝑆)       (𝜑 → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))
 
Theoremrescabs2 17094 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐶𝑉)    &   (𝜑𝐽 Fn (𝑇 × 𝑇))    &   (𝜑𝑆𝑊)    &   (𝜑𝑇𝑆)       (𝜑 → ((𝐶s 𝑆) ↾cat 𝐽) = (𝐶cat 𝐽))
 
Theoremissubc 17095* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐻 = (Homf𝐶)    &    1 = (Id‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑆 = dom dom 𝐽)       (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
 
Theoremissubc2 17096* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐻 = (Homf𝐶)    &    1 = (Id‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐽 Fn (𝑆 × 𝑆))       (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
 
Theorem0ssc 17097 For any category 𝐶, the empty set is a subcategory subset of 𝐶. (Contributed by AV, 23-Apr-2020.)
(𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))
 
Theorem0subcat 17098 For any category 𝐶, the empty set is a (full) subcategory of 𝐶, see example 4.3(1.a) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)
(𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))
 
Theoremcatsubcat 17099 For any category 𝐶, 𝐶 itself is a (full) subcategory of 𝐶, see example 4.3(1.b) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)
(𝐶 ∈ Cat → (Homf𝐶) ∈ (Subcat‘𝐶))
 
Theoremsubcssc 17100 An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   𝐻 = (Homf𝐶)       (𝜑𝐽cat 𝐻)
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