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Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhomfeq 16401* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐵 = (Base‘𝐷))       (𝜑 → ((Homf𝐶) = (Homf𝐷) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦)))
 
Theoremhomfeqd 16402 If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (Base‘𝐶) = (Base‘𝐷))    &   (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))       (𝜑 → (Homf𝐶) = (Homf𝐷))
 
Theoremhomfeqbas 16403 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → (Base‘𝐶) = (Base‘𝐷))
 
Theoremhomfeqval 16404 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌))
 
Theoremcomfffval 16405* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
 
Theoremcomffval 16406* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
 
Theoremcomfval 16407 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
 
Theoremcomfffval2 16408* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
 
Theoremcomffval2 16409* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌· 𝑍)𝑓)))
 
Theoremcomfval2 16410 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
 
Theoremcomfffn 16411 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)       𝑂 Fn ((𝐵 × 𝐵) × 𝐵)
 
Theoremcomffn 16412 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))
 
Theoremcomfeq 16413* Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
· = (comp‘𝐶)    &    = (comp‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐵 = (Base‘𝐶))    &   (𝜑𝐵 = (Base‘𝐷))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → ((compf𝐶) = (compf𝐷) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦 𝑧)𝑓)))
 
Theoremcomfeqd 16414 Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑 → (comp‘𝐶) = (comp‘𝐷))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))       (𝜑 → (compf𝐶) = (compf𝐷))
 
Theoremcomfeqval 16415 Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
 
Theoremcatpropd 16416 Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
 
Theoremcidpropd 16417 Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (Id‘𝐶) = (Id‘𝐷))
 
8.1.2  Opposite category
 
Syntaxcoppc 16418 The opposite category operation.
class oppCat
 
Definitiondf-oppc 16419* Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. Definition 3.5 of [Adamek] p. 25. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑓)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑓)(1st𝑢)))⟩))
 
Theoremoppcval 16420* Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)       (𝐶𝑉𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
 
Theoremoppchomfval 16421 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝑂 = (oppCat‘𝐶)       tpos 𝐻 = (Hom ‘𝑂)
 
Theoremoppchom 16422 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐻 = (Hom ‘𝐶)    &   𝑂 = (oppCat‘𝐶)       (𝑋(Hom ‘𝑂)𝑌) = (𝑌𝐻𝑋)
 
Theoremoppccofval 16423 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌· 𝑋))
 
Theoremoppcco 16424 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐶)    &    · = (comp‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍)𝐹) = (𝐹(⟨𝑍, 𝑌· 𝑋)𝐺))
 
Theoremoppcbas 16425 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Base‘𝐶)       𝐵 = (Base‘𝑂)
 
Theoremoppccatid 16426 Lemma for oppccat 16429. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝑂 = (oppCat‘𝐶)       (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶)))
 
Theoremoppchomf 16427 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐻 = (Homf𝐶)       tpos 𝐻 = (Homf𝑂)
 
Theoremoppcid 16428 Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Id‘𝐶)       (𝐶 ∈ Cat → (Id‘𝑂) = 𝐵)
 
Theoremoppccat 16429 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑂 = (oppCat‘𝐶)       (𝐶 ∈ Cat → 𝑂 ∈ Cat)
 
Theorem2oppcbas 16430 The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 16444. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Base‘𝐶)       𝐵 = (Base‘(oppCat‘𝑂))
 
Theorem2oppchomf 16431 The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 16444. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝑂 = (oppCat‘𝐶)       (Homf𝐶) = (Homf ‘(oppCat‘𝑂))
 
Theorem2oppccomf 16432 The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 16444. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝑂 = (oppCat‘𝐶)       (compf𝐶) = (compf‘(oppCat‘𝑂))
 
Theoremoppchomfpropd 16433 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 16434 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 16435 Extend class notation with the class of all monomorphisms.
class Mono
 
Syntaxcepi 16436 Extend class notation with the class of all epimorphisms.
class Epi
 
Definitiondf-mon 16437* 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 16438 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 16439* 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 16440* 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 16441* 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 16442 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 16443 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 16444 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 16445 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 16446 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 16447* 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 16448* 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 16449 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 16450 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 16451 Extend class notation with the sections of a morphism.
class Sect
 
Syntaxcinv 16452 Extend class notation with the inverses of a morphism.
class Inv
 
Syntaxciso 16453 Extend class notation with the class of all isomorphisms.
class Iso
 
Definitiondf-sect 16454* 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 16455* 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 16456* 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 16457* 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 16458* 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 16459 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 16460 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 16461 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 16462 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 16463 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 16464* 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 16465* 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 16466 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 16467 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 16468 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 16469 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 16470 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 16471 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 16472 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 16473 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 16474 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 16475 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 16476 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 16477 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 16478 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 16479* 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 16480* 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 16481 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 16482 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 16483 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 16484 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 16485 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 16486 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 16487 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 16488 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 16489 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 16490 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 16491 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 16492 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 16493 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 16494 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 16495 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 16496 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 16497 The inverse of an isomorphism 𝐹 (which is unique because of invf 16475 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 16498 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 16499 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 16500 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))
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