P. 181 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	natcl 18001	A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)𝐽(𝐾‘𝑋)))
Theorem	natfn 18002	A natural transformation is a function on the objects of 𝐶. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝜑 → 𝐴 Fn 𝐵)
Theorem	nati 18003	Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩))    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ · = (comp‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌))    ⇒   ⊢ (𝜑 → ((𝐴‘𝑌)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ · (𝐾‘𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹‘𝑋), (𝐾‘𝑋)⟩ · (𝐾‘𝑌))(𝐴‘𝑋)))
Theorem	wunnat 18004	A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
Theorem	catstr 18005	A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} Struct ⟨1, ;15⟩
Theorem	fucval 18006*	Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝐵 = (𝐶 Func 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ · = (comp‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ (𝜑 → ∙ = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))    ⇒   ⊢ (𝜑 → 𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})
Theorem	fuccofval 18007*	Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝐵 = (𝐶 Func 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ · = (comp‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ ∙ = (comp‘𝑄)    ⇒   ⊢ (𝜑 → ∙ = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
Theorem	fucbas 18008	The objects of the functor category are functors from 𝐶 to 𝐷. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    ⇒   ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
Theorem	fuchom 18009	The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    ⇒   ⊢ 𝑁 = (Hom ‘𝑄)
Theorem	fucco 18010*	Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ · = (comp‘𝐷)    &   ⊢ ∙ = (comp‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))    ⇒   ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
Theorem	fuccoval 18011	Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 𝐴 = (Base‘𝐶)    &   ⊢ · = (comp‘𝐷)    &   ⊢ ∙ = (comp‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑋) = ((𝑆‘𝑋)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)⟩ · ((1st ‘𝐻)‘𝑋))(𝑅‘𝑋)))
Theorem	fuccocl 18012	The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ ∙ = (comp‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))    ⇒   ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
Theorem	fucidcl 18013	The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → ( 1 ∘ (1st ‘𝐹)) ∈ (𝐹𝑁𝐹))
Theorem	fuclid 18014	Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ ∙ = (comp‘𝑄)    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))    ⇒   ⊢ (𝜑 → (( 1 ∘ (1st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = 𝑅)
Theorem	fucrid 18015	Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ ∙ = (comp‘𝑄)    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))    ⇒   ⊢ (𝜑 → (𝑅(⟨𝐹, 𝐹⟩ ∙ 𝐺)( 1 ∘ (1st ‘𝐹))) = 𝑅)
Theorem	fucass 18016	Associativity of natural transformation composition. Remark 6.14(b) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ ∙ = (comp‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))    &   ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ (𝐻𝑁𝐾))    ⇒   ⊢ (𝜑 → ((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)(⟨𝐹, 𝐺⟩ ∙ 𝐾)𝑅) = (𝑇(⟨𝐹, 𝐻⟩ ∙ 𝐾)(𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)))
Theorem	fuccatid 18017*	The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    &   ⊢ 1 = (Id‘𝐷)    ⇒   ⊢ (𝜑 → (𝑄 ∈ Cat ∧ (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1st ‘𝑓)))))
Theorem	fuccat 18018	The functor category is a category. Remark 6.16 in [Adamek] p. 88. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝑄 ∈ Cat)
Theorem	fucid 18019	The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝐼 = (Id‘𝑄)    &   ⊢ 1 = (Id‘𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    ⇒   ⊢ (𝜑 → (𝐼‘𝐹) = ( 1 ∘ (1st ‘𝐹)))
Theorem	fucsect 18020*	Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   ⊢ 𝑆 = (Sect‘𝑄)    &   ⊢ 𝑇 = (Sect‘𝐷)    ⇒   ⊢ (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1st ‘𝐹)‘𝑥)𝑇((1st ‘𝐺)‘𝑥))(𝑉‘𝑥))))
Theorem	fucinv 18021*	Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   ⊢ 𝐼 = (Inv‘𝑄)    &   ⊢ 𝐽 = (Inv‘𝐷)    ⇒   ⊢ (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1st ‘𝐹)‘𝑥)𝐽((1st ‘𝐺)‘𝑥))(𝑉‘𝑥))))
Theorem	invfuc 18022*	If 𝑉(𝑥) is an inverse to 𝑈(𝑥) for each 𝑥, and 𝑈 is a natural transformation, then 𝑉 is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   ⊢ 𝐼 = (Inv‘𝑄)    &   ⊢ 𝐽 = (Inv‘𝐷)    &   ⊢ (𝜑 → 𝑈 ∈ (𝐹𝑁𝐺))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥)(((1st ‘𝐹)‘𝑥)𝐽((1st ‘𝐺)‘𝑥))𝑋)    ⇒   ⊢ (𝜑 → 𝑈(𝐹𝐼𝐺)(𝑥 ∈ 𝐵 ↦ 𝑋))
Theorem	fuciso 18023*	A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
⊢ 𝑄 = (𝐶 FuncCat 𝐷)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝑁 = (𝐶 Nat 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   ⊢ 𝐼 = (Iso‘𝑄)    &   ⊢ 𝐽 = (Iso‘𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1st ‘𝐹)‘𝑥)𝐽((1st ‘𝐺)‘𝑥)))))
Theorem	natpropd 18024	If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ Cat)    &   ⊢ (𝜑 → 𝐵 ∈ Cat)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
Theorem	fucpropd 18025	If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))    &   ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))    &   ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))    &   ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))    &   ⊢ (𝜑 → 𝐴 ∈ Cat)    &   ⊢ (𝜑 → 𝐵 ∈ Cat)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐷 ∈ Cat)    ⇒   ⊢ (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))
8.1.10  Initial, terminal and zero objects of a category
Syntax	cinito 18026	Extend class notation with the class of initial objects of a category.
class InitO
Syntax	ctermo 18027	Extend class notation with the class of terminal objects of a category.
class TermO
Syntax	czeroo 18028	Extend class notation with the class of zero objects of a category.
class ZeroO
Definition	df-inito 18029*	An object A is said to be an initial object provided that for each object B there is exactly one morphism from A to B. Definition 7.1 in [Adamek] p. 101, or definition in [Lang] p. 57 (called "a universally repelling object" there). See dfinito2 18048 and dfinito3 18050 for alternate definitions depending on df-termo 18030. (Contributed by AV, 3-Apr-2020.)
⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
Definition	df-termo 18030*	An object A is called a terminal object provided that for each object B there is exactly one morphism from B to A. Definition 7.4 in [Adamek] p. 102, or definition in [Lang] p. 57 (called "a universally attracting object" there). See dftermo2 18049 and dftermo3 18051 for alternate definitions depending on df-inito 18029. (Contributed by AV, 3-Apr-2020.)
⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
Definition	df-zeroo 18031	An object A is called a zero object provided that it is both an initial object and a terminal object. Definition 7.7 of [Adamek] p. 103. (Contributed by AV, 3-Apr-2020.)
⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
Theorem	initofn 18032	InitO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ InitO Fn Cat
Theorem	termofn 18033	TermO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ TermO Fn Cat
Theorem	zeroofn 18034	ZeroO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ ZeroO Fn Cat
Theorem	initorcl 18035	Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020.)
⊢ (𝐼 ∈ (InitO‘𝐶) → 𝐶 ∈ Cat)
Theorem	termorcl 18036	Reverse closure for a terminal object: If a class has a terminal object, the class is a category. (Contributed by AV, 4-Apr-2020.)
⊢ (𝑇 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)
Theorem	zeroorcl 18037	Reverse closure for a zero object: If a class has a zero object, the class is a category. (Contributed by AV, 4-Apr-2020.)
⊢ (𝑍 ∈ (ZeroO‘𝐶) → 𝐶 ∈ Cat)
Theorem	initoval 18038*	The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})
Theorem	termoval 18039*	The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (TermO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑎)})
Theorem	zerooval 18040	The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))
Theorem	isinito 18041*	The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏)))
Theorem	istermo 18042*	The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝐼)))
Theorem	iszeroo 18043	The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶))))
Theorem	isinitoi 18044*	Implication of a class being an initial object. (Contributed by AV, 6-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏)))
Theorem	istermoi 18045*	Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑏𝐻𝑂)))
Theorem	initoid 18046	For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 6-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})
Theorem	termoid 18047	For a terminal object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 18-Apr-2020.)
⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ ((𝜑 ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})
Theorem	dfinito2 18048	An initial object is a terminal object in the opposite category. An alternate definition of df-inito 18029 depending on df-termo 18030. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ InitO = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐)))
Theorem	dftermo2 18049	A terminal object is an initial object in the opposite category. An alternate definition of df-termo 18030 depending on df-inito 18029. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ TermO = (𝑐 ∈ Cat ↦ (InitO‘(oppCat‘𝑐)))
Theorem	dfinito3 18050	An alternate definition of df-inito 18029 depending on df-termo 18030, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ InitO = (TermO ∘ (oppCat ↾ Cat))
Theorem	dftermo3 18051	An alternate definition of df-termo 18030 depending on df-inito 18029, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
⊢ TermO = (InitO ∘ (oppCat ↾ Cat))
Theorem	initoo 18052	An initial object is an object. (Contributed by AV, 14-Apr-2020.)
⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
Theorem	termoo 18053	A terminal object is an object. (Contributed by AV, 18-Apr-2020.)
⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
Theorem	iszeroi 18054	Implication of a class being a zero object. (Contributed by AV, 18-Apr-2020.)
⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (ZeroO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ (𝑂 ∈ (InitO‘𝐶) ∧ 𝑂 ∈ (TermO‘𝐶))))
Theorem	2initoinv 18055	Morphisms between two initial objects are inverses. (Contributed by AV, 14-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))    &   ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))    ⇒   ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)
Theorem	initoeu1 18056*	Initial objects are essentially unique (strong form), i.e. there is a unique isomorphism between two initial objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 14-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))    &   ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
Theorem	initoeu1w 18057	Initial objects are essentially unique (weak form), i.e. if A and B are initial objects, then A and B are isomorphic. Proposition 7.3 (1) of [Adamek] p. 102. (Contributed by AV, 6-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))    &   ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))    ⇒   ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)
Theorem	initoeu2lem0 18058	Lemma 0 for initoeu2 18061. (Contributed by AV, 9-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ ⚬ = (comp‘𝐶)    ⇒   ⊢ (((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) ∧ ((𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾)) = (𝐺(⟨𝐴, 𝐵⟩ ⚬ 𝐷)((𝐵(Inv‘𝐶)𝐴)‘𝐾))) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾))
Theorem	initoeu2lem1 18059*	Lemma 1 for initoeu2 18061. (Contributed by AV, 9-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ ⚬ = (comp‘𝐶)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐺 = (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾)))
Theorem	initoeu2lem2 18060*	Lemma 2 for initoeu2 18061. (Contributed by AV, 10-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))    &   ⊢ 𝑋 = (Base‘𝐶)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ 𝐼 = (Iso‘𝐶)    &   ⊢ ⚬ = (comp‘𝐶)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴⟩ ⚬ 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) → ∃!𝑔 𝑔 ∈ (𝐵𝐻𝐷)))
Theorem	initoeu2 18061	Initial objects are essentially unique, if A is an initial object, then so is every object that is isomorphic to A. Proposition 7.3 (2) in [Adamek] p. 102. (Contributed by AV, 10-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))    &   ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))
Theorem	2termoinv 18062	Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))    &   ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))    ⇒   ⊢ ((𝜑 ∧ 𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)
Theorem	termoeu1 18063*	Terminal objects are essentially unique (strong form), i.e. there is a unique isomorphism between two terminal objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 18-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))    &   ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
Theorem	termoeu1w 18064	Terminal objects are essentially unique (weak form), i.e. if A and B are terminal objects, then A and B are isomorphic. Proposition 7.6 of [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))    &   ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))    ⇒   ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)
8.2  Arrows (disjointified hom-sets)
Syntax	cdoma 18065	Extend class notation to include the domain extractor for an arrow.
class doma
Syntax	ccoda 18066	Extend class notation to include the codomain extractor for an arrow.
class coda
Syntax	carw 18067	Extend class notation to include the collection of all arrows of a category.
class Arrow
Syntax	choma 18068	Extend class notation to include the set of all arrows with a specific domain and codomain.
class Homa
Definition	df-doma 18069	Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
⊢ doma = (1st ∘ 1st )
Definition	df-coda 18070	Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
⊢ coda = (2nd ∘ 1st )
Definition	df-homa 18071*	Definition of the hom-set extractor for arrows, which tags the morphisms of the underlying hom-set with domain and codomain, which can then be extracted using df-doma 18069 and df-coda 18070. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
Definition	df-arw 18072	Definition of the set of arrows of a category. We will use the term "arrow" to denote a morphism tagged with its domain and codomain, as opposed to Hom, which allows hom-sets for distinct objects to overlap. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐))
Theorem	homarcl 18073	Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
Theorem	homafval 18074*	Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐽 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽‘𝑥))))
Theorem	homaf 18075	Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    ⇒   ⊢ (𝜑 → 𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))
Theorem	homaval 18076	Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐽 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
Theorem	elhoma 18077	Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐽 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
Theorem	elhomai 18078	Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐽 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))    ⇒   ⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)
Theorem	elhomai2 18079	Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ 𝐽 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))    ⇒   ⊢ (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))
Theorem	homarcl2 18080	Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
Theorem	homarel 18081	An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ Rel (𝑋𝐻𝑌)
Theorem	homa1 18082	The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝑍 = ⟨𝑋, 𝑌⟩)
Theorem	homahom2 18083	The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐶)    ⇒   ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌))
Theorem	homahom 18084	The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋𝐽𝑌))
Theorem	homadm 18085	The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = 𝑋)
Theorem	homacd 18086	The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌)
Theorem	homadmcd 18087	Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd ‘𝐹)⟩)
Theorem	arwval 18088	The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ 𝐴 = ∪ ran 𝐻
Theorem	arwrcl 18089	The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat)
Theorem	arwhoma 18090	An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹)))
Theorem	homarw 18091	A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐻 = (Homa‘𝐶)    ⇒   ⊢ (𝑋𝐻𝑌) ⊆ 𝐴
Theorem	arwdm 18092	The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (doma‘𝐹) ∈ 𝐵)
Theorem	arwcd 18093	The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (coda‘𝐹) ∈ 𝐵)
Theorem	dmaf 18094	The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (doma ↾ 𝐴):𝐴⟶𝐵
Theorem	cdaf 18095	The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (coda ↾ 𝐴):𝐴⟶𝐵
Theorem	arwhom 18096	The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    &   ⊢ 𝐽 = (Hom ‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (2nd ‘𝐹) ∈ ((doma‘𝐹)𝐽(coda‘𝐹)))
Theorem	arwdmcd 18097	Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
⊢ 𝐴 = (Arrow‘𝐶)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹 = ⟨(doma‘𝐹), (coda‘𝐹), (2nd ‘𝐹)⟩)
8.2.1  Identity and composition for arrows
Syntax	cida 18098	Extend class notation to include identity for arrows.
class Ida
Syntax	ccoa 18099	Extend class notation to include composition for arrows.
class compa
Definition	df-ida 18100*	Definition of the identity arrow, which is just the identity morphism tagged with its domain and codomain. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))