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Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-sect 16401* 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 16402* 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 16403* 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 16404* 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 16405* 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 16406 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 16407 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 16408 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 16409 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 16410 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 16411* 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 16412* 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 16413 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 16414 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 16415 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 16416 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 16417 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 16418 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 16419 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 16420 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 16421 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 16422 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 16423 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 16424 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 16425 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 16426* Alternate definition of an isomorphism of a category, according to definition 3.8 in [Adamek] p. 28. (Contributed by AV, 10-Apr-2017.)
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 16427* Alternate definition of an isomorphism of a category as a section in both directions. (Contributed by AV, 11-Apr-2017.)
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 16428 If there are two inverses of an morphism, these inverses are equal. Corollary 3.11 of [Adamek] p. 28. (Contributed by AV, 10-Apr-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))

Theoremisofn 16429 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-2017.)
(𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Theoremisohom 16430 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 16431 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 16432 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 16433 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 16434 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 16435 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 16436 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 16437 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 16438 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 16439 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 16440 The identity is a section of itself. (Contributed by AV, 8-Apr-2017.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋)(𝑋(Sect‘𝐶)𝑋)(𝐼𝑋))

Theoreminvid 16441 The inverse of the identity is the identity. (Contributed by AV, 8-Apr-2017.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋)(𝑋(Inv‘𝐶)𝑋)(𝐼𝑋))

Theoremidiso 16442 The identity is an isomorphism. Example 3.13 of [Adamek] p. 28. (Contributed by AV, 8-Apr-2017.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋) ∈ (𝑋(Iso‘𝐶)𝑋))

Theoremidinv 16443 The inverse of the identity is the identity. Example 3.13 of [Adamek] p. 28. (Contributed by AV, 9-Apr-2017.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ((𝑋(Inv‘𝐶)𝑋)‘(𝐼𝑋)) = (𝐼𝑋))

Theoreminvisoinvl 16444 The inverse of an isomorphism 𝐹 (which is unique because of invf 16422 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-2017.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))       (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)

Theoreminvisoinvr 16445 The inverse of an isomorphism is invers to the isomorphism. (Contributed by AV, 9-Apr-2017.)
𝐵 = (Base‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   𝑁 = (Inv‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐼𝑌))       (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))

Theoreminvcoisoid 16446 The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 5-Apr-2017.)
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 16447 The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-Apr-2017.)
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 16448 Right cancellation of an inverse of an isomorphism. (Contributed by AV, 5-Apr-2017.)
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 16450). It is shown that this relation is an equivalence relation, see cicer 16460.

Syntaxccic 16449 Extend class notation to include the category isomorphism relation.
class 𝑐

Definitiondf-cic 16450 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 17696. (Contributed by AV, 4-Apr-2020.)
𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))

Theoremcicfval 16451 The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.)
(𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))

Theorembrcic 16452 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 16453* 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 16454 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 16455 Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐𝐶)𝑂)

Theoremciclcl 16456 Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))

Theoremcicrcl 16457 Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))

Theoremcicsym 16458 Isomorphism is symmetric. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆( ≃𝑐𝐶)𝑅)

Theoremcictr 16459 Isomorphism is transitive. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆𝑆( ≃𝑐𝐶)𝑇) → 𝑅( ≃𝑐𝐶)𝑇)

Theoremcicer 16460 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 16461 Extend class notation to include the subset relation for subcategories.
class cat

Syntaxcresc 16462 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
class cat

Syntaxcsubc 16463 Extend class notation to include the collection of subcategories of a category.
class Subcat

Definitiondf-ssc 16464* 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 16466, 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 16465* 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 16466* (Subcat‘𝐶) is the set of all the subcategory specifications of the category 𝐶. Like df-subg 17585, 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 16465) 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 16467 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Rel ⊆cat

Theorembrssc 16468* 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 16469* An analogue of pwex 4846 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 16470 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)

Theoremsscfn1 16471 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 16472 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 16473 Lemma for ssc1 16475 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻 Fn (𝑆 × 𝑆))       (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V))

Theoremisssc 16474* 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 16475 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 16476 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 16477 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 16478 The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝐻cat 𝐻)

Theoremssctr 16479 The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)

Theoremssceq 16480 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)

Theoremrescval 16481 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 16482 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 16483 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 16484 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 16485 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 16486 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 16487 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 16488 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 16489* 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 16490* 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 16491 For any category 𝐶, the empty set is a subcategory subset of 𝐶. (Contributed by AV, 23-Apr-2020.)
(𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))

Theorem0subcat 16492 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 16493 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 16494 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)

Theoremsubcfn 16495 An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝑆 = dom dom 𝐽)       (𝜑𝐽 Fn (𝑆 × 𝑆))

Theoremsubcss1 16496 The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &   𝐵 = (Base‘𝐶)       (𝜑𝑆𝐵)

Theoremsubcss2 16497 The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Theoremsubcidcl 16498 The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &   (𝜑𝑋𝑆)    &    1 = (Id‘𝐶)       (𝜑 → ( 1𝑋) ∈ (𝑋𝐽𝑋))

Theoremsubccocl 16499 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &   (𝜑𝑋𝑆)    &    · = (comp‘𝐶)    &   (𝜑𝑌𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐹 ∈ (𝑋𝐽𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐽𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍))

Theoremsubccatid 16500* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &    1 = (Id‘𝐶)       (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑥𝑆 ↦ ( 1𝑥))))

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