HomeHome Metamath Proof Explorer
Theorem List (p. 162 of 425)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26947)
  Hilbert Space Explorer  Hilbert Space Explorer
(26948-28472)
  Users' Mathboxes  Users' Mathboxes
(28473-42426)
 

Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoppchomfpropd 16101 If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷)))
 
Theoremoppccomfpropd 16102 If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))       (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷)))
 
8.1.3  Monomorphisms and epimorphisms
 
Syntaxcmon 16103 Extend class notation with the class of all monomorphisms.
class Mono
 
Syntaxcepi 16104 Extend class notation with the class of all epimorphisms.
class Epi
 
Definitiondf-mon 16105* Function returning the monomorphisms of the category 𝑐. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
Mono = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (𝑥𝑏, 𝑦𝑏 ↦ {𝑓 ∈ (𝑥𝑦) ∣ ∀𝑧𝑏 Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}))
 
Definitiondf-epi 16106 Function returning the epimorphisms of the category 𝑐. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
 
Theoremmonfval 16107* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝑀 = (Mono‘𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑𝑀 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
 
Theoremismon 16108* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝑀 = (Mono‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋· 𝑌)𝑔)))))
 
Theoremismon2 16109* Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝑀 = (Mono‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋· 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋· 𝑌)) → 𝑔 = ))))
 
Theoremmonhom 16110 A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝑀 = (Mono‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌))
 
Theoremmoni 16111 Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝑀 = (Mono‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝑀𝑌))    &   (𝜑𝐺 ∈ (𝑍𝐻𝑋))    &   (𝜑𝐾 ∈ (𝑍𝐻𝑋))       (𝜑 → ((𝐹(⟨𝑍, 𝑋· 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋· 𝑌)𝐾) ↔ 𝐺 = 𝐾))
 
Theoremmonpropd 16112 If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (Mono‘𝐶) = (Mono‘𝐷))
 
Theoremoppcmon 16113 A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝑀 = (Mono‘𝑂)    &   𝐸 = (Epi‘𝐶)       (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))
 
Theoremoppcepi 16114 An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐸 = (Epi‘𝑂)    &   𝑀 = (Mono‘𝐶)       (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋))
 
Theoremisepi 16115* Definition of an epimorphism in a category. (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) ∧ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(⟨𝑋, 𝑌· 𝑧)𝐹)))))
 
Theoremisepi2 16116* 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 16117 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 16118 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 16119 Extend class notation with the sections of a morphism.
class Sect
 
Syntaxcinv 16120 Extend class notation with the inverses of a morphism.
class Inv
 
Syntaxciso 16121 Extend class notation with the class of all isomorphisms.
class Iso
 
Definitiondf-sect 16122* 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 16123* 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 16124* 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 16125* 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 16126* 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 16127 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 16128 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 16129 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 16130 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 16131 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 16132* 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 16133* 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 16134 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 16135 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 16136 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 16137 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 16138 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 16139 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 16140 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 16141 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 16142 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 16143 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 16144 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 16145 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 16146 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 16147* 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 16148* 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 16149 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 16150 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 16151 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 16152 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 16153 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 16154 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 16155 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 16156 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 16157 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 16158 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 16159 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 16160 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 16161 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 16162 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 16163 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 16164 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 16165 The inverse of an isomorphism 𝐹 (which is unique because of invf 16143 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 16166 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 16167 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 16168 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 16169 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 16171). It is shown that this relation is an equivalence relation, see cicer 16181.

 
Syntaxccic 16170 Extend class notation to include the category isomorphism relation.
class 𝑐
 
Definitiondf-cic 16171 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 17417. (Contributed by AV, 4-Apr-2020.)
𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
 
Theoremcicfval 16172 The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.)
(𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
 
Theorembrcic 16173 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 16174* 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 16175 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 16176 Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐𝐶)𝑂)
 
Theoremciclcl 16177 Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
 
Theoremcicrcl 16178 Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
 
Theoremcicsym 16179 Isomorphism is symmetric. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆( ≃𝑐𝐶)𝑅)
 
Theoremcictr 16180 Isomorphism is transitive. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆𝑆( ≃𝑐𝐶)𝑇) → 𝑅( ≃𝑐𝐶)𝑇)
 
Theoremcicer 16181 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 16182 Extend class notation to include the subset relation for subcategories.
class cat
 
Syntaxcresc 16183 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
class cat
 
Syntaxcsubc 16184 Extend class notation to include the collection of subcategories of a category.
class Subcat
 
Definitiondf-ssc 16185* 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 16187, 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 16186* 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 16187* (Subcat‘𝐶) is the set of all the subcategory specifications of the category 𝐶. Like df-subg 17306, 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 16186) 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 16188 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Rel ⊆cat
 
Theorembrssc 16189* 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 16190* An analogue of pwex 4673 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 16191 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
 
Theoremsscfn1 16192 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 16193 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 16194 Lemma for ssc1 16196 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻 Fn (𝑆 × 𝑆))       (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V))
 
Theoremisssc 16195* 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 16196 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 16197 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 16198 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 16199 The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝐻cat 𝐻)
 
Theoremssctr 16200 The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
  Copyright terms: Public domain < Previous  Next >