P. 169 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	idinv 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)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋(Inv‘𝐶)𝑋)‘(𝐼‘𝑋)) = (𝐼‘𝑋))
Theorem	invisoinvl 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)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
Theorem	invisoinvr 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)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
Theorem	invcoisoid 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)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)    ⇒   ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))
Theorem	isocoinvid 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)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ ⚬ = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌))
Theorem	rcaninv 16806	Right cancellation of an inverse of an isomorphism. (Contributed by AV, 5-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (Inv‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))    &   ⊢ (𝜑 → 𝐻 ∈ (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.
Syntax	ccic 16807	Extend class notation to include the category isomorphism relation.
class ≃𝑐
Definition	df-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 ∅))
Theorem	cicfval 16809	The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.)
⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))
Theorem	brcic 16810	The relation "is isomorphic to" for categories. (Contributed by AV, 5-Apr-2020.)
⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ (𝑋𝐼𝑌) ≠ ∅))
Theorem	cic 16811*	Objects 𝑋 and 𝑌 in a category are isomorphic provided that there is an isomorphism 𝑓:𝑋⟶𝑌, see definition 3.15 of [Adamek] p. 29. (Contributed by AV, 4-Apr-2020.)
⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋𝐼𝑌)))
Theorem	brcici 16812	Prove that two objects are isomorphic by an explicit isomorphism. (Contributed by AV, 4-Apr-2020.)
⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌)
Theorem	cicref 16813	Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)
⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐 ‘𝐶)𝑂)
Theorem	ciclcl 16814	Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.)
⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
Theorem	cicrcl 16815	Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.)
⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
Theorem	cicsym 16816	Isomorphism is symmetric. (Contributed by AV, 5-Apr-2020.)
⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆( ≃𝑐 ‘𝐶)𝑅)
Theorem	cictr 16817	Isomorphism is transitive. (Contributed by AV, 5-Apr-2020.)
⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆 ∧ 𝑆( ≃𝑐 ‘𝐶)𝑇) → 𝑅( ≃𝑐 ‘𝐶)𝑇)
Theorem	cicer 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 → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶))
8.1.6  Subcategories
Syntax	cssc 16819	Extend class notation to include the subset relation for subcategories.
class ⊆cat
Syntax	cresc 16820	Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
class ↾cat
Syntax	csubc 16821	Extend class notation to include the collection of subcategories of a category.
class Subcat
Definition	df-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 = {⟨ℎ, 𝑗⟩ ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))}
Definition	df-resc 16823*	Define the restriction of a category to a given set of arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
Definition	df-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 ↦ {ℎ ∣ (ℎ ⊆cat (Homf ‘𝑐) ∧ [dom dom ℎ / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥ℎ𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥ℎ𝑧)))})
Theorem	sscrel 16825	The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ Rel ⊆cat
Theorem	brssc 16826*	The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
Theorem	sscpwex 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.)
⊢ {ℎ ∣ ℎ ⊆cat 𝐽} ∈ V
Theorem	subcrcl 16828	Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
Theorem	sscfn1 16829	The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐻 ⊆cat 𝐽)    &   ⊢ (𝜑 → 𝑆 = dom dom 𝐻)    ⇒   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
Theorem	sscfn2 16830	The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐻 ⊆cat 𝐽)    &   ⊢ (𝜑 → 𝑇 = dom dom 𝐽)    ⇒   ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))
Theorem	ssclem 16831	Lemma for ssc1 16833 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    ⇒   ⊢ (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V))
Theorem	isssc 16832*	Value of the subcategory subset relation when the arguments are known functions. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐻 ⊆cat 𝐽 ↔ (𝑆 ⊆ 𝑇 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
Theorem	ssc1 16833	Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))    &   ⊢ (𝜑 → 𝐻 ⊆cat 𝐽)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
Theorem	ssc2 16834	Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝐻 ⊆cat 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
Theorem	sscres 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) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)
Theorem	sscid 16836	The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝐻 ⊆cat 𝐻)
Theorem	ssctr 16837	The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐶) → 𝐴 ⊆cat 𝐶)
Theorem	ssceq 16838	The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ ((𝐴 ⊆cat 𝐵 ∧ 𝐵 ⊆cat 𝐴) → 𝐴 = 𝐵)
Theorem	rescval 16839	Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐻)    ⇒   ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Theorem	rescval2 16840	Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐻)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    ⇒   ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Theorem	rescbas 16841	Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐻)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
Theorem	reschom 16842	Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐻)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐻 = (Hom ‘𝐷))
Theorem	reschomf 16843	Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐻)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐻 = (Homf ‘𝐷))
Theorem	rescco 16844	Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐻)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (comp‘𝐷))
Theorem	rescabs 16845	Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
Theorem	rescabs2 16846	Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐶 ↾s 𝑆) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
Theorem	issubc 16847*	Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑆 = dom dom 𝐽)    ⇒   ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
Theorem	issubc2 16848*	Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐻 = (Homf ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    ⇒   ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
Theorem	0ssc 16849	For any category 𝐶, the empty set is a subcategory subset of 𝐶. (Contributed by AV, 23-Apr-2020.)
⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))
Theorem	0subcat 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))
Theorem	catsubcat 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 → (Homf ‘𝐶) ∈ (Subcat‘𝐶))
Theorem	subcssc 16852	An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ 𝐻 = (Homf ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐽 ⊆cat 𝐻)
Theorem	subcfn 16853	An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝑆 = dom dom 𝐽)    ⇒   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
Theorem	subcss1 16854	The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
Theorem	subcss2 16855	The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))
Theorem	subcidcl 16856	The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))
Theorem	subccocl 16857	A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐽𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍))
Theorem	subccatid 16858*	A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    &   ⊢ 1 = (Id‘𝐶)    ⇒   ⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑥 ∈ 𝑆 ↦ ( 1 ‘𝑥))))
Theorem	subcid 16859	The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = ((Id‘𝐷)‘𝑋))
Theorem	subccat 16860	A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    ⇒   ⊢ (𝜑 → 𝐷 ∈ Cat)
Theorem	issubc3 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 = (Homf ‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ 𝐷 = (𝐶 ↾cat 𝐽)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))    ⇒   ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ 𝐷 ∈ Cat)))
Theorem	fullsubc 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 = (Homf ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))
Theorem	fullresc 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 = (Homf ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ 𝐷 = (𝐶 ↾s 𝑆)    &   ⊢ 𝐸 = (𝐶 ↾cat (𝐻 ↾ (𝑆 × 𝑆)))    ⇒   ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸)))
Theorem	resscat 16864	A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ ((𝐶 ∈ Cat ∧ 𝑆 ∈ 𝑉) → (𝐶 ↾s 𝑆) ∈ Cat)
Theorem	subsubc 16865	A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾cat 𝐻)    ⇒   ⊢ (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽 ⊆cat 𝐻)))
8.1.7  Functors
Syntax	cfunc 16866	Extend class notation with the class of all functors.
class Func
Syntax	cidfu 16867	Extend class notation with identity functor.
class idfunc
Syntax	ccofu 16868	Extend class notation with functor composition.
class ∘func
Syntax	cresf 16869	Extend class notation to include restriction of a functor to a subcategory.
class ↾f
Definition	df-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) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑢)(𝑓‘(2nd ‘𝑧))) ↑𝑚 ((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(⟨(𝑓‘𝑥), (𝑓‘𝑦)⟩(comp‘𝑢)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))})
Definition	df-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)))⟩)
Definition	df-cofu 16872*	Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ ⟨((1st ‘𝑔) ∘ (1st ‘𝑓)), (𝑥 ∈ dom dom (2nd ‘𝑓), 𝑦 ∈ dom dom (2nd ‘𝑓) ↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))⟩)
Definition	df-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, ℎ ∈ V ↦ ⟨((1st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩)
Theorem	relfunc 16874	The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ Rel (𝐷 Func 𝐸)
Theorem	funcrcl 16875	Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝐹 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
Theorem	isfunc 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)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
Theorem	isfuncd 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)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → 𝐸 ∈ Cat)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))    ⇒   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
Theorem	funcf1 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)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
Theorem	funcixp 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)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))
Theorem	funcf2 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)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	funcfn2 16881	The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐷)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    ⇒   ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
Theorem	funcid 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)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝐹‘𝑋)))
Theorem	funcco 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)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝐺𝑍)‘𝑁)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝑀)))
Theorem	funcsect 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)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀(𝑋𝑆𝑌)𝑁)    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝑇(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))
Theorem	funcinv 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)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀(𝑋𝐼𝑌)𝑁)    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))
Theorem	funciso 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)    &   ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌))    ⇒   ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
Theorem	funcoppc 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)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺)
Theorem	idfuval 16888*	Value of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐼 = ⟨( I ↾ 𝐵), (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧)))⟩)
Theorem	idfu2nd 16889	Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ (𝑋𝐻𝑌)))
Theorem	idfu2 16890	Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝑋(2nd ‘𝐼)𝑌)‘𝐹) = 𝐹)
Theorem	idfu1st 16891	Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵))
Theorem	idfu1 16892	Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋)
Theorem	idfucl 16893	The identity functor is a functor. Example 3.20(1) of [Adamek] p. 30. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐼 = (idfunc‘𝐶)    ⇒   ⊢ (𝐶 ∈ Cat → 𝐼 ∈ (𝐶 Func 𝐶))
Theorem	cofuval 16894*	Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
Theorem	cofu1st 16895	Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐹)) = ((1st ‘𝐺) ∘ (1st ‘𝐹)))
Theorem	cofu1 16896	Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((1st ‘(𝐺 ∘func 𝐹))‘𝑋) = ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)))
Theorem	cofu2nd 16897	Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))
Theorem	cofu2 16898	Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌)‘𝑅) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)))
Theorem	cofuval2 16899*	Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    &   ⊢ (𝜑 → 𝐻(𝐷 Func 𝐸)𝐾)    ⇒   ⊢ (𝜑 → (⟨𝐻, 𝐾⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨(𝐻 ∘ 𝐹), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ∘ (𝑥𝐺𝑦)))⟩)
Theorem	cofucl 16900	The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))    ⇒   ⊢ (𝜑 → (𝐺 ∘func 𝐹) ∈ (𝐶 Func 𝐸))