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Theorem List for Metamath Proof Explorer - 16301-16400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfthinv 16301 A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀 ∈ (𝑋𝐻𝑌))    &   (𝜑𝑁 ∈ (𝑌𝐻𝑋))    &   𝐼 = (Inv‘𝐶)    &   𝐽 = (Inv‘𝐷)       (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹𝑋)𝐽(𝐹𝑌))((𝑌𝐺𝑋)‘𝑁)))

Theoremfthmon 16302 A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))    &   𝑀 = (Mono‘𝐶)    &   𝑁 = (Mono‘𝐷)    &   (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))       (𝜑𝑅 ∈ (𝑋𝑀𝑌))

Theoremfthepi 16303 A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))    &   𝐸 = (Epi‘𝐶)    &   𝑃 = (Epi‘𝐷)    &   (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑃(𝐹𝑌)))       (𝜑𝑅 ∈ (𝑋𝐸𝑌))

Theoremffthiso 16304 A fully faithful functor reflects isomorphisms. Corollary 3.32 of [Adamek] p. 35. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹(𝐶 Faith 𝐷)𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐹(𝐶 Full 𝐷)𝐺)    &   𝐼 = (Iso‘𝐶)    &   𝐽 = (Iso‘𝐷)       (𝜑 → (𝑅 ∈ (𝑋𝐼𝑌) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))))

Theoremfthres2b 16305* Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑅 ∈ (Subcat‘𝐷))    &   (𝜑𝑅 Fn (𝑆 × 𝑆))    &   (𝜑𝐹:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝐺𝑦):𝑌⟶((𝐹𝑥)𝑅(𝐹𝑦)))       (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Faith (𝐷cat 𝑅))𝐺))

Theoremfthres2c 16306 Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐸 = (𝐷s 𝑆)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝐴𝑆)       (𝜑 → (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Faith 𝐸)𝐺))

Theoremfthres2 16307 A faithful functor into a restricted category is also a faithful functor into the whole category. (Contributed by Mario Carneiro, 27-Jan-2017.)
(𝑅 ∈ (Subcat‘𝐷) → (𝐶 Faith (𝐷cat 𝑅)) ⊆ (𝐶 Faith 𝐷))

Theoremidffth 16308 The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐼 = (idfunc𝐶)       (𝐶 ∈ Cat → 𝐼 ∈ ((𝐶 Full 𝐶) ∩ (𝐶 Faith 𝐶)))

Theoremcofull 16309 The composition of two full functors is full. Proposition 3.30(d) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Full 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Full 𝐸))       (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Full 𝐸))

Theoremcofth 16310 The composition of two faithful functors is faithful. Proposition 3.30(c) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
(𝜑𝐹 ∈ (𝐶 Faith 𝐷))    &   (𝜑𝐺 ∈ (𝐷 Faith 𝐸))       (𝜑 → (𝐺func 𝐹) ∈ (𝐶 Faith 𝐸))

Theoremcoffth 16311 The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
(𝜑𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)))    &   (𝜑𝐺 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))       (𝜑 → (𝐺func 𝐹) ∈ ((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸)))

Theoremrescfth 16312 The inclusion functor from a subcategory is a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐷 = (𝐶cat 𝐽)    &   𝐼 = (idfunc𝐷)       (𝐽 ∈ (Subcat‘𝐶) → 𝐼 ∈ (𝐷 Faith 𝐶))

Theoremressffth 16313 The inclusion functor from a full subcategory is a full and faithful functor, see also remark 4.4(2) in [Adamek] p. 49. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐷 = (𝐶s 𝑆)    &   𝐼 = (idfunc𝐷)       ((𝐶 ∈ Cat ∧ 𝑆𝑉) → 𝐼 ∈ ((𝐷 Full 𝐶) ∩ (𝐷 Faith 𝐶)))

Theoremfullres2c 16314 Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐸 = (𝐷s 𝑆)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝐴𝑆)       (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Full 𝐸)𝐺))

Theoremffthres2c 16315 Condition for a fully faithful functor to also be a fully faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
𝐴 = (Base‘𝐶)    &   𝐸 = (𝐷s 𝑆)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝐴𝑆)       (𝜑 → (𝐹((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝐺𝐹((𝐶 Full 𝐸) ∩ (𝐶 Faith 𝐸))𝐺))

8.1.9  Natural transformations and the functor category

Syntaxcnat 16316 Extend class notation to include the collection of natural transformations.
class Nat

Syntaxcfuc 16317 Extend class notation to include the functor category.
class FuncCat

Definitiondf-nat 16318* Definition of a natural transformation between two functors. A natural transformation 𝐴:𝐹𝐺 is a collection of arrows 𝐴(𝑥):𝐹(𝑥)⟶𝐺(𝑥), such that 𝐴(𝑦) ∘ 𝐹() = 𝐺() ∘ 𝐴(𝑥) for each morphism :𝑥𝑦. Definition 6.1 in [Adamek] p. 83, and definition in [Lang] p. 65. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥))}))

Definitiondf-fuc 16319* Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. Definition 6.15 in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))))⟩})

Theoremfnfuc 16320 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
FuncCat Fn (Cat × Cat)

Theoremnatfval 16321* Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)       𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})

Theoremisnat 16322* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)    &   (𝜑𝐹(𝐶 Func 𝐷)𝐺)    &   (𝜑𝐾(𝐶 Func 𝐷)𝐿)       (𝜑 → (𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩) ↔ (𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨(𝐹𝑥), (𝐹𝑦)⟩ · (𝐾𝑦))((𝑥𝐺𝑦)‘)) = (((𝑥𝐿𝑦)‘)(⟨(𝐹𝑥), (𝐾𝑥)⟩ · (𝐾𝑦))(𝐴𝑥)))))

Theoremisnat2 16323* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))       (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴X𝑥𝐵 (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝐴𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩ · ((1st𝐺)‘𝑦))((𝑥(2nd𝐹)𝑦)‘)) = (((𝑥(2nd𝐺)𝑦)‘)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐺)‘𝑦))(𝐴𝑥)))))

Theoremnatffn 16324 The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)       𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))

Theoremnatrcl 16325 Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)       (𝐴 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))

Theoremnat1st2nd 16326 Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (𝐹𝑁𝐺))       (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))

Theoremnatixp 16327* A natural transformation is a function from the objects of 𝐶 to homomorphisms from 𝐹(𝑥) to 𝐺(𝑥). (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))    &   𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)       (𝜑𝐴X𝑥𝐵 ((𝐹𝑥)𝐽(𝐾𝑥)))

Theoremnatcl 16328 A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))    &   𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐴𝑋) ∈ ((𝐹𝑋)𝐽(𝐾𝑋)))

Theoremnatfn 16329 A natural transformation is a function on the objects of 𝐶. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))    &   𝐵 = (Base‘𝐶)       (𝜑𝐴 Fn 𝐵)

Theoremnati 16330 Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐴 ∈ (⟨𝐹, 𝐺𝑁𝐾, 𝐿⟩))    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐷)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝐴𝑌)(⟨(𝐹𝑋), (𝐹𝑌)⟩ · (𝐾𝑌))((𝑋𝐺𝑌)‘𝑅)) = (((𝑋𝐿𝑌)‘𝑅)(⟨(𝐹𝑋), (𝐾𝑋)⟩ · (𝐾𝑌))(𝐴𝑋)))

Theoremwunnat 16331 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐶𝑈)    &   (𝜑𝐷𝑈)       (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)

Theoremcatstr 16332 A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.)
{⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} Struct ⟨1, 15⟩

Theoremfucval 16333* 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑 = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))       (𝜑𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})

Theoremfuccofval 16334* 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &    = (comp‘𝑄)       (𝜑 = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))

Theoremfucbas 16335 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)

Theoremfuchom 16336 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)       𝑁 = (Hom ‘𝑄)

Theoremfucco 16337* 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)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))    &   (𝜑𝑆 ∈ (𝐺𝑁𝐻))       (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) = (𝑥𝐴 ↦ ((𝑆𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ · ((1st𝐻)‘𝑥))(𝑅𝑥))))

Theoremfuccoval 16338 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)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))    &   (𝜑𝑆 ∈ (𝐺𝑁𝐻))    &   (𝜑𝑋𝐴)       (𝜑 → ((𝑆(⟨𝐹, 𝐺 𝐻)𝑅)‘𝑋) = ((𝑆𝑋)(⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩ · ((1st𝐻)‘𝑋))(𝑅𝑋)))

Theoremfuccocl 16339 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)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))    &   (𝜑𝑆 ∈ (𝐺𝑁𝐻))       (𝜑 → (𝑆(⟨𝐹, 𝐺 𝐻)𝑅) ∈ (𝐹𝑁𝐻))

Theoremfucidcl 16340 The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &    1 = (Id‘𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))       (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))

Theoremfuclid 16341 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)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))       (𝜑 → (( 1 ∘ (1st𝐺))(⟨𝐹, 𝐺 𝐺)𝑅) = 𝑅)

Theoremfucrid 16342 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)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))       (𝜑 → (𝑅(⟨𝐹, 𝐹 𝐺)( 1 ∘ (1st𝐹))) = 𝑅)

Theoremfucass 16343 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)    &   (𝜑𝑅 ∈ (𝐹𝑁𝐺))    &   (𝜑𝑆 ∈ (𝐺𝑁𝐻))    &   (𝜑𝑇 ∈ (𝐻𝑁𝐾))       (𝜑 → ((𝑇(⟨𝐺, 𝐻 𝐾)𝑆)(⟨𝐹, 𝐺 𝐾)𝑅) = (𝑇(⟨𝐹, 𝐻 𝐾)(𝑆(⟨𝐹, 𝐺 𝐻)𝑅)))

Theoremfuccatid 16344* The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &    1 = (Id‘𝐷)       (𝜑 → (𝑄 ∈ Cat ∧ (Id‘𝑄) = (𝑓 ∈ (𝐶 Func 𝐷) ↦ ( 1 ∘ (1st𝑓)))))

Theoremfuccat 16345 The functor category is a category. Remark 6.16 in [Adamek] p. 88. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑𝑄 ∈ Cat)

Theoremfucid 16346 The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝐼 = (Id‘𝑄)    &    1 = (Id‘𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))       (𝜑 → (𝐼𝐹) = ( 1 ∘ (1st𝐹)))

Theoremfucsect 16347* 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)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   𝑆 = (Sect‘𝑄)    &   𝑇 = (Sect‘𝐷)       (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝑇((1st𝐺)‘𝑥))(𝑉𝑥))))

Theoremfucinv 16348* 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)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   𝐼 = (Inv‘𝑄)    &   𝐽 = (Inv‘𝐷)       (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥))))

Theoreminvfuc 16349* 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)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   𝐼 = (Inv‘𝑄)    &   𝐽 = (Inv‘𝐷)    &   (𝜑𝑈 ∈ (𝐹𝑁𝐺))    &   ((𝜑𝑥𝐵) → (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))𝑋)       (𝜑𝑈(𝐹𝐼𝐺)(𝑥𝐵𝑋))

Theoremfuciso 16350* 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)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   𝐼 = (Iso‘𝑄)    &   𝐽 = (Iso‘𝐷)       (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))))

Theoremnatpropd 16351 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 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴 ∈ Cat)    &   (𝜑𝐵 ∈ Cat)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))

Theoremfucpropd 16352 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 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴 ∈ Cat)    &   (𝜑𝐵 ∈ Cat)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))

8.1.10  Initial, terminal and zero objects of a category

Syntaxcinito 16353 Extend class notation with the class of initial objects of a category.
class InitO

Syntaxctermo 16354 Extend class notation with the class of terminal objects of a category.
class TermO

Syntaxczeroo 16355 Extend class notation with the class of zero objects of a category.
class ZeroO

Definitiondf-inito 16356* 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). (Contributed by AV, 3-Apr-2020.)
InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑎(Hom ‘𝑐)𝑏)})

Definitiondf-termo 16357* 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). (Contributed by AV, 3-Apr-2020.)
TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)})

Definitiondf-zeroo 16358 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)))

Theoreminitorcl 16359 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)

Theoremtermorcl 16360 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)

Theoremzeroorcl 16361 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)

Theoreminitoval 16362* The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (InitO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑎𝐻𝑏)})

Theoremtermoval 16363* The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})

Theoremzerooval 16364 The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶)))

Theoremisinito 16365* The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐼𝐵)       (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝐼𝐻𝑏)))

Theoremistermo 16366* The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐼𝐵)       (𝜑 → (𝐼 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝐼)))

Theoremiszeroo 16367 The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐼𝐵)       (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶))))

Theoremisinitoi 16368* Implication of a class being an initial object. (Contributed by AV, 6-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)       ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑂𝐻𝑏)))

Theoremistermoi 16369* Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐶 ∈ Cat)       ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))

Theoreminitoid 16370 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)    &   (𝜑𝐶 ∈ Cat)       ((𝜑𝑂 ∈ (InitO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Theoremtermoid 16371 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)    &   (𝜑𝐶 ∈ Cat)       ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐻𝑂) = {((Id‘𝐶)‘𝑂)})

Theoreminitoo 16372 An initial object is an object. (Contributed by AV, 14-Apr-2020.)
(𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Theoremtermoo 16373 A terminal object is an object. (Contributed by AV, 18-Apr-2020.)
(𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Theoremiszeroi 16374 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))))

Theorem2initoinv 16375 Morphisms between two initial objects are inverses. (Contributed by AV, 14-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   (𝜑𝐵 ∈ (InitO‘𝐶))       ((𝜑𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Theoreminitoeu1 16376* 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.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   (𝜑𝐵 ∈ (InitO‘𝐶))       (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))

Theoreminitoeu1w 16377 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.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   (𝜑𝐵 ∈ (InitO‘𝐶))       (𝜑𝐴( ≃𝑐𝐶)𝐵)

Theoreminitoeu2lem0 16378 Lemma 0 for initoeu2 16381. (Contributed by AV, 9-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   𝑋 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &    = (comp‘𝐶)       (((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋)) ∧ (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))

Theoreminitoeu2lem1 16379* Lemma 1 for initoeu2 16381. (Contributed by AV, 9-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   𝑋 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &    = (comp‘𝐶)       ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → ((∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ 𝐺 ∈ (𝐵𝐻𝐷)) → 𝐺 = (𝐹(⟨𝐵, 𝐴 𝐷)𝐾)))

Theoreminitoeu2lem2 16380* Lemma 2 for initoeu2 16381. (Contributed by AV, 10-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   𝑋 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &    = (comp‘𝐶)       ((𝜑 ∧ (𝐴𝑋𝐵𝑋𝐷𝑋) ∧ (𝐾 ∈ (𝐵𝐼𝐴) ∧ 𝐹 ∈ (𝐴𝐻𝐷) ∧ (𝐹(⟨𝐵, 𝐴 𝐷)𝐾) ∈ (𝐵𝐻𝐷))) → (∃!𝑓 𝑓 ∈ (𝐴𝐻𝐷) → ∃!𝑔 𝑔 ∈ (𝐵𝐻𝐷)))

Theoreminitoeu2 16381 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.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (InitO‘𝐶))    &   (𝜑𝐴( ≃𝑐𝐶)𝐵)       (𝜑𝐵 ∈ (InitO‘𝐶))

Theorem2termoinv 16382 Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (TermO‘𝐶))    &   (𝜑𝐵 ∈ (TermO‘𝐶))       ((𝜑𝐺 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝐹 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐹(𝐴(Inv‘𝐶)𝐵)𝐺)

Theoremtermoeu1 16383* 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.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (TermO‘𝐶))    &   (𝜑𝐵 ∈ (TermO‘𝐶))       (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))

Theoremtermoeu1w 16384 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.)
(𝜑𝐶 ∈ Cat)    &   (𝜑𝐴 ∈ (TermO‘𝐶))    &   (𝜑𝐵 ∈ (TermO‘𝐶))       (𝜑𝐴( ≃𝑐𝐶)𝐵)

8.2  Arrows (disjointified hom-sets)

Syntaxcdoma 16385 Extend class notation to include the domain extractor for an arrow.
class doma

Syntaxccoda 16386 Extend class notation to include the codomain extractor for an arrow.
class coda

Syntaxcarw 16387 Extend class notation to include the collection of all arrows of a category.
class Arrow

Syntaxchoma 16388 Extend class notation to include the set of all arrows with a specific domain and codomain.
class Homa

Definitiondf-doma 16389 Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
doma = (1st ∘ 1st )

Definitiondf-coda 16390 Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
coda = (2nd ∘ 1st )

Definitiondf-homa 16391* 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 16389 and df-coda 16390. (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))))

Definitiondf-arw 16392 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 (Homa𝑐))

Theoremhomarcl 16393 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Theoremhomafval 16394* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥 ∈ (𝐵 × 𝐵) ↦ ({𝑥} × (𝐽𝑥))))

Theoremhomaf 16395 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑𝐻:(𝐵 × 𝐵)⟶𝒫 ((𝐵 × 𝐵) × V))

Theoremhomaval 16396 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))

Theoremelhoma 16397 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))

Theoremelhomai 16398 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐽𝑌))       (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)

Theoremelhomai2 16399 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐽 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐽𝑌))       (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))

Theoremhomarcl2 16400 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋𝐵𝑌𝐵))

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