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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwunnat 17001 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 17002 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 17003* 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 17004* 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 17005 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 17006 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 17007* 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 17008 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 17009 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 17010 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 17011 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 17012 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 17013 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 17014* 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 17015 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 17016 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 17017* 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 17018* 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 17019* 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 17020* 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 17021 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 17022 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 17023 Extend class notation with the class of initial objects of a category.
class InitO
 
Syntaxctermo 17024 Extend class notation with the class of terminal objects of a category.
class TermO
 
Syntaxczeroo 17025 Extend class notation with the class of zero objects of a category.
class ZeroO
 
Definitiondf-inito 17026* 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 17027* 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 17028 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 17029 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 17030 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 17031 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 17032* 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 17033* 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 17034 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 17035* 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 17036* 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 17037 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 17038* 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 17039* 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 17040 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 17041 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 17042 An initial object is an object. (Contributed by AV, 14-Apr-2020.)
(𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
 
Theoremtermoo 17043 A terminal object is an object. (Contributed by AV, 18-Apr-2020.)
(𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
 
Theoremiszeroi 17044 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 17045 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 17046* 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 17047 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 17048 Lemma 0 for initoeu2 17051. (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 17049* Lemma 1 for initoeu2 17051. (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 17050* Lemma 2 for initoeu2 17051. (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 17051 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 17052 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 17053* 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 17054 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 17055 Extend class notation to include the domain extractor for an arrow.
class doma
 
Syntaxccoda 17056 Extend class notation to include the codomain extractor for an arrow.
class coda
 
Syntaxcarw 17057 Extend class notation to include the collection of all arrows of a category.
class Arrow
 
Syntaxchoma 17058 Extend class notation to include the set of all arrows with a specific domain and codomain.
class Homa
 
Definitiondf-doma 17059 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 17060 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 17061* 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 17059 and df-coda 17060. (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 17062 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 17063 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
 
Theoremhomafval 17064* 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 17065 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 17066 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 17067 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 17068 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 17069 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 17070 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))
 
Theoremhomarel 17071 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       Rel (𝑋𝐻𝑌)
 
Theoremhoma1 17072 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝑍(𝑋𝐻𝑌)𝐹𝑍 = ⟨𝑋, 𝑌⟩)
 
Theoremhomahom2 17073 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐽 = (Hom ‘𝐶)       (𝑍(𝑋𝐻𝑌)𝐹𝐹 ∈ (𝑋𝐽𝑌))
 
Theoremhomahom 17074 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &   𝐽 = (Hom ‘𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (2nd𝐹) ∈ (𝑋𝐽𝑌))
 
Theoremhomadm 17075 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
 
Theoremhomacd 17076 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
 
Theoremhomadmcd 17077 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)
 
Theoremarwval 17078 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐻 = (Homa𝐶)       𝐴 = ran 𝐻
 
Theoremarwrcl 17079 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)       (𝐹𝐴𝐶 ∈ Cat)
 
Theoremarwhoma 17080 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 = (Homa𝐶)       (𝐹𝐴𝐹 ∈ ((doma𝐹)𝐻(coda𝐹)))
 
Theoremhomarw 17081 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐻 = (Homa𝐶)       (𝑋𝐻𝑌) ⊆ 𝐴
 
Theoremarwdm 17082 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹𝐴 → (doma𝐹) ∈ 𝐵)
 
Theoremarwcd 17083 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (𝐹𝐴 → (coda𝐹) ∈ 𝐵)
 
Theoremdmaf 17084 The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (doma𝐴):𝐴𝐵
 
Theoremcdaf 17085 The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)    &   𝐵 = (Base‘𝐶)       (coda𝐴):𝐴𝐵
 
Theoremarwhom 17086 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)       (𝐹𝐴 → (2nd𝐹) ∈ ((doma𝐹)𝐽(coda𝐹)))
 
Theoremarwdmcd 17087 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐴 = (Arrow‘𝐶)       (𝐹𝐴𝐹 = ⟨(doma𝐹), (coda𝐹), (2nd𝐹)⟩)
 
8.2.1  Identity and composition for arrows
 
Syntaxcida 17088 Extend class notation to include identity for arrows.
class Ida
 
Syntaxccoa 17089 Extend class notation to include composition for arrows.
class compa
 
Definitiondf-ida 17090* 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)⟩))
 
Definitiondf-coa 17091* Definition of the composition of arrows. Since arrows are tagged with domain and codomain, this does not need to be a quinary operation like the regular composition in a category comp. Instead, it is a partial binary operation on arrows, which is defined when the domain of the first arrow matches the codomain of the second. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩))
 
Theoremidafval 17092* Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)       (𝜑𝐼 = (𝑥𝐵 ↦ ⟨𝑥, 𝑥, ( 1𝑥)⟩))
 
Theoremidaval 17093 Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋) = ⟨𝑋, 𝑋, ( 1𝑋)⟩)
 
Theoremida2 17094 Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &    1 = (Id‘𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → (2nd ‘(𝐼𝑋)) = ( 1𝑋))
 
Theoremidahom 17095 Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝐻 = (Homa𝐶)       (𝜑 → (𝐼𝑋) ∈ (𝑋𝐻𝑋))
 
Theoremidadm 17096 Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (doma‘(𝐼𝑋)) = 𝑋)
 
Theoremidacd 17097 Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)       (𝜑 → (coda‘(𝐼𝑋)) = 𝑋)
 
Theoremidaf 17098 The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐼 = (Ida𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐴 = (Arrow‘𝐶)       (𝜑𝐼:𝐵𝐴)
 
Theoremcoafval 17099* The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐴 = (Arrow‘𝐶)    &    = (comp‘𝐶)        · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
 
Theoremeldmcoa 17100 A pair 𝐺, 𝐹 is in the domain of the arrow composition, if the domain of 𝐺 equals the codomain of 𝐹. (In this case we say 𝐺 and 𝐹 are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐴 = (Arrow‘𝐶)       (𝐺dom · 𝐹 ↔ (𝐹𝐴𝐺𝐴 ∧ (coda𝐹) = (doma𝐺)))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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