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Theorem List for Metamath Proof Explorer - 16801-16900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremidinv 16801 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 16802 The inverse of an isomorphism 𝐹 (which is unique because of invf 16780 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 16803 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 16804 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 16805 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 16806 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 16808). It is shown that this relation is an equivalence relation, see cicer 16818.

Syntaxccic 16807 Extend class notation to include the category isomorphism relation.
class 𝑐

Definitiondf-cic 16808 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 18053. (Contributed by AV, 4-Apr-2020.)
𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))

Theoremcicfval 16809 The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.)
(𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))

Theorembrcic 16810 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 16811* 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 16812 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 16813 Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐𝐶)𝑂)

Theoremciclcl 16814 Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))

Theoremcicrcl 16815 Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))

Theoremcicsym 16816 Isomorphism is symmetric. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆( ≃𝑐𝐶)𝑅)

Theoremcictr 16817 Isomorphism is transitive. (Contributed by AV, 5-Apr-2020.)
((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆𝑆( ≃𝑐𝐶)𝑇) → 𝑅( ≃𝑐𝐶)𝑇)

Theoremcicer 16818 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 16819 Extend class notation to include the subset relation for subcategories.
class cat

Syntaxcresc 16820 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
class cat

Syntaxcsubc 16821 Extend class notation to include the collection of subcategories of a category.
class Subcat

Definitiondf-ssc 16822* 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 16824, 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 16823* 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 16824* (Subcat‘𝐶) is the set of all the subcategory specifications of the category 𝐶. Like df-subg 17942, 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 16823) 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 16825 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Rel ⊆cat

Theorembrssc 16826* 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 16827* An analogue of pwex 5080 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 16828 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)

Theoremsscfn1 16829 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 16830 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 16831 Lemma for ssc1 16833 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝜑𝐻 Fn (𝑆 × 𝑆))       (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V))

Theoremisssc 16832* 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 16833 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 16834 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 16835 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 16836 The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝐻cat 𝐻)

Theoremssctr 16837 The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐴cat 𝐵𝐵cat 𝐶) → 𝐴cat 𝐶)

Theoremssceq 16838 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐴cat 𝐵𝐵cat 𝐴) → 𝐴 = 𝐵)

Theoremrescval 16839 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 16840 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 16841 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 16842 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 16843 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 16844 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 16845 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 16846 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 16847* 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 16848* 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 16849 For any category 𝐶, the empty set is a subcategory subset of 𝐶. (Contributed by AV, 23-Apr-2020.)
(𝐶 ∈ Cat → ∅ ⊆cat (Homf𝐶))

Theorem0subcat 16850 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 16851 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 16852 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 16853 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 16854 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 16855 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 16856 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 16857 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 16858* 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))))

Theoremsubcid 16859 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐽 ∈ (Subcat‘𝐶))    &   (𝜑𝐽 Fn (𝑆 × 𝑆))    &    1 = (Id‘𝐶)    &   (𝜑𝑋𝑆)       (𝜑 → ( 1𝑋) = ((Id‘𝐷)‘𝑋))

Theoremsubccat 16860 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐽 ∈ (Subcat‘𝐶))       (𝜑𝐷 ∈ Cat)

Theoremissubc3 16861* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 17701, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐻 = (Homf𝐶)    &    1 = (Id‘𝐶)    &   𝐷 = (𝐶cat 𝐽)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐽 Fn (𝑆 × 𝑆))       (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))

Theoremfullsubc 16862 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory), see definition 4.1(2) of [Adamek] p. 48. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑆𝐵)       (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))

Theoremfullresc 16863 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Homf𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑆𝐵)    &   𝐷 = (𝐶s 𝑆)    &   𝐸 = (𝐶cat (𝐻 ↾ (𝑆 × 𝑆)))       (𝜑 → ((Homf𝐷) = (Homf𝐸) ∧ (compf𝐷) = (compf𝐸)))

Theoremresscat 16864 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
((𝐶 ∈ Cat ∧ 𝑆𝑉) → (𝐶s 𝑆) ∈ Cat)

Theoremsubsubc 16865 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐷 = (𝐶cat 𝐻)       (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻)))

8.1.7  Functors

Syntaxcfunc 16866 Extend class notation with the class of all functors.
class Func

Syntaxcidfu 16867 Extend class notation with identity functor.
class idfunc

Syntaxccofu 16868 Extend class notation with functor composition.
class func

Syntaxcresf 16869 Extend class notation to include restriction of a functor to a subcategory.
class f

Definitiondf-func 16870* Function returning all the functors from a category 𝑡 to a category 𝑢. Definition 3.17 of [Adamek] p. 29, and definition in [Lang] p. 62 ("covariant functor"). Intuitively a functor associates any morphism of 𝑡 to a morphism of 𝑢, any object of 𝑡 to an object of 𝑢, and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of 𝑡 to an object of 𝑢 we write it associates any identity of 𝑡 to an identity of 𝑢 which simplifies the definition. According to remark 3.19 in [Adamek] p. 30, "a functor F : A -> B is technically a family of functions; one from Ob(A) to Ob(B) [here: f, called "the object part" in the following], and for each pair (A,A') of A-objects, one from hom(A,A') to hom(FA, FA') [here: g, called "the morphism part" in the following]". (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st𝑧))(Hom ‘𝑢)(𝑓‘(2nd𝑧))) ↑𝑚 ((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓𝑥)) ∧ ∀𝑦𝑏𝑧𝑏𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓𝑥), (𝑓𝑦)⟩(comp‘𝑢)(𝑓𝑧))((𝑥𝑔𝑦)‘𝑚))))})

Definitiondf-idfu 16871* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc = (𝑡 ∈ Cat ↦ (Base‘𝑡) / 𝑏⟨( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))⟩)

Definitiondf-cofu 16872* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st𝑔) ∘ (1st𝑓)), (𝑥 ∈ dom dom (2nd𝑓), 𝑦 ∈ dom dom (2nd𝑓) ↦ ((((1st𝑓)‘𝑥)(2nd𝑔)((1st𝑓)‘𝑦)) ∘ (𝑥(2nd𝑓)𝑦)))⟩)

Definitiondf-resf 16873* Define the restriction of a functor to a subcategory (analogue of df-res 5354). (Contributed by Mario Carneiro, 6-Jan-2017.)
f = (𝑓 ∈ V, ∈ V ↦ ⟨((1st𝑓) ↾ dom dom ), (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥)))⟩)

Theoremrelfunc 16874 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Rel (𝐷 Func 𝐸)

Theoremfuncrcl 16875 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
(𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))

Theoremisfunc 16876* Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &    1 = (Id‘𝐷)    &   𝐼 = (Id‘𝐸)    &    · = (comp‘𝐷)    &   𝑂 = (comp‘𝐸)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)       (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵𝐶𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))

Theoremisfuncd 16877* Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &    1 = (Id‘𝐷)    &   𝐼 = (Id‘𝐸)    &    · = (comp‘𝐷)    &   𝑂 = (comp‘𝐸)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)    &   (𝜑𝐹:𝐵𝐶)    &   (𝜑𝐺 Fn (𝐵 × 𝐵))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))    &   ((𝜑𝑥𝐵) → ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))       (𝜑𝐹(𝐷 Func 𝐸)𝐺)

Theoremfuncf1 16878 The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)       (𝜑𝐹:𝐵𝐶)

Theoremfuncixp 16879* The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)       (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))

Theoremfuncf2 16880 The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))

Theoremfuncfn2 16881 The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐷)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)       (𝜑𝐺 Fn (𝐵 × 𝐵))

Theoremfuncid 16882 A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &    1 = (Id‘𝐷)    &   𝐼 = (Id‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)       (𝜑 → ((𝑋𝐺𝑋)‘( 1𝑋)) = (𝐼‘(𝐹𝑋)))

Theoremfuncco 16883 A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐷)    &    · = (comp‘𝐷)    &   𝑂 = (comp‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑀 ∈ (𝑋𝐻𝑌))    &   (𝜑𝑁 ∈ (𝑌𝐻𝑍))       (𝜑 → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌· 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹𝑋), (𝐹𝑌)⟩𝑂(𝐹𝑍))((𝑋𝐺𝑌)‘𝑀)))

Theoremfuncsect 16884 The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝑆 = (Sect‘𝐷)    &   𝑇 = (Sect‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀(𝑋𝑆𝑌)𝑁)       (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝑇(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))

Theoremfuncinv 16885 The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐼 = (Inv‘𝐷)    &   𝐽 = (Inv‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀(𝑋𝐼𝑌)𝑁)       (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁))

Theoremfunciso 16886 The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐷)    &   𝐼 = (Iso‘𝐷)    &   𝐽 = (Iso‘𝐸)    &   (𝜑𝐹(𝐷 Func 𝐸)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀 ∈ (𝑋𝐼𝑌))       (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))

Theoremfuncoppc 16887 A functor on categories yields a functor on the opposite categories (in the same direction), see definition 3.41 of [Adamek] p. 39. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑃 = (oppCat‘𝐷)    &   (𝜑𝐹(𝐶 Func 𝐷)𝐺)       (𝜑𝐹(𝑂 Func 𝑃)tpos 𝐺)

Theoremidfuval 16888* Value of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻𝑧)))⟩)

Theoremidfu2nd 16889 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋(2nd𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))

Theoremidfu2 16890 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝑋(2nd𝐼)𝑌)‘𝐹) = 𝐹)

Theoremidfu1st 16891 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑 → (1st𝐼) = ( I ↾ 𝐵))

Theoremidfu1 16892 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → ((1st𝐼)‘𝑋) = 𝑋)

Theoremidfucl 16893 The identity functor is a functor. Example 3.20(1) of [Adamek] p. 30. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐼 = (idfunc𝐶)       (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))

Theoremcofuval 16894* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))       (𝜑 → (𝐺func 𝐹) = ⟨((1st𝐺) ∘ (1st𝐹)), (𝑥𝐵, 𝑦𝐵 ↦ ((((1st𝐹)‘𝑥)(2nd𝐺)((1st𝐹)‘𝑦)) ∘ (𝑥(2nd𝐹)𝑦)))⟩)

Theoremcofu1st 16895 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))       (𝜑 → (1st ‘(𝐺func 𝐹)) = ((1st𝐺) ∘ (1st𝐹)))

Theoremcofu1 16896 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))    &   (𝜑𝑋𝐵)       (𝜑 → ((1st ‘(𝐺func 𝐹))‘𝑋) = ((1st𝐺)‘((1st𝐹)‘𝑋)))

Theoremcofu2nd 16897 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋(2nd ‘(𝐺func 𝐹))𝑌) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌)) ∘ (𝑋(2nd𝐹)𝑌)))

Theoremcofu2 16898 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝑋(2nd ‘(𝐺func 𝐹))𝑌)‘𝑅) = ((((1st𝐹)‘𝑋)(2nd𝐺)((1st𝐹)‘𝑌))‘((𝑋(2nd𝐹)𝑌)‘𝑅)))

Theoremcofuval2 16899* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐵 = (Base‘𝐶)    &   (𝜑𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑𝐻(𝐷 Func 𝐸)𝐾)       (𝜑 → (⟨𝐻, 𝐾⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐻𝐹), (𝑥𝐵, 𝑦𝐵 ↦ (((𝐹𝑥)𝐾(𝐹𝑦)) ∘ (𝑥𝐺𝑦)))⟩)

Theoremcofucl 16900 The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Func 𝐸))       (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Func 𝐸))

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