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Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremistermoi 16701* 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 16702 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 16703 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 16704 An initial object is an object. (Contributed by AV, 14-Apr-2020.)
(𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Theoremtermoo 16705 A terminal object is an object. (Contributed by AV, 18-Apr-2020.)
(𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Theoremiszeroi 16706 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 16707 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 16708* 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 16709 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 16710 Lemma 0 for initoeu2 16713. (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 16711* Lemma 1 for initoeu2 16713. (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 16712* Lemma 2 for initoeu2 16713. (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 16713 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 16714 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 16715* 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 16716 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 16717 Extend class notation to include the domain extractor for an arrow.
class doma

Syntaxccoda 16718 Extend class notation to include the codomain extractor for an arrow.
class coda

Syntaxcarw 16719 Extend class notation to include the collection of all arrows of a category.
class Arrow

Syntaxchoma 16720 Extend class notation to include the set of all arrows with a specific domain and codomain.
class Homa

Definitiondf-doma 16721 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 16722 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 16723* 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 16721 and df-coda 16722. (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 16724 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 16725 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Theoremhomafval 16726* 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 16727 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 16728 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 16729 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 16730 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 16731 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 16732 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 16733 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       Rel (𝑋𝐻𝑌)

Theoremhoma1 16734 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 16735 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 16736 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 16737 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)

Theoremhomacd 16738 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)

Theoremhomadmcd 16739 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)       (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2nd𝐹)⟩)

Theoremarwval 16740 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 16741 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 16742 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 16743 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 16744 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 16745 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 16746 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 16747 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 16748 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 16749 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 16750 Extend class notation to include identity for arrows.
class Ida

Syntaxccoa 16751 Extend class notation to include composition for arrows.
class compa

Definitiondf-ida 16752* 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 16753* 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 16754* 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 16755 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 16756 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 16757 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 16758 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 16759 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 16760 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 16761* 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 16762 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)))

Theoremdmcoass 16763 The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐴 = (Arrow‘𝐶)       dom · ⊆ (𝐴 × 𝐴)

Theoremhomdmcoa 16764 If 𝐹:𝑋𝑌 and 𝐺:𝑌𝑍, then 𝐺 and 𝐹 are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐻 = (Homa𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑𝐺dom · 𝐹)

Theoremcoaval 16765 Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐻 = (Homa𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &    = (comp‘𝐶)       (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)

Theoremcoa2 16766 The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐻 = (Homa𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &    = (comp‘𝐶)       (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))

Theoremcoahom 16767 The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐻 = (Homa𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))       (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

Theoremcoapm 16768 Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
· = (compa𝐶)    &   𝐴 = (Arrow‘𝐶)        · ∈ (𝐴pm (𝐴 × 𝐴))

Theoremarwlid 16769 Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &    · = (compa𝐶)    &    1 = (Ida𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (( 1𝑌) · 𝐹) = 𝐹)

Theoremarwrid 16770 Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &    · = (compa𝐶)    &    1 = (Ida𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → (𝐹 · ( 1𝑋)) = 𝐹)

Theoremarwass 16771 Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝐻 = (Homa𝐶)    &    · = (compa𝐶)    &    1 = (Ida𝐶)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑍))    &   (𝜑𝐾 ∈ (𝑍𝐻𝑊))       (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))

8.3  Examples of categories

8.3.1  The category of sets

Syntaxcsetc 16772 Extend class notation to include the category Set.
class SetCat

Definitiondf-setc 16773* Definition of the category Set, relativized to a subset 𝑢. Example 3.3(1) of [Adamek] p. 22. This is the category of all sets in 𝑢 and functions between these sets. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.)
SetCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦𝑚 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩})

Theoremsetcval 16774* Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦𝑚 𝑥)))    &   (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremsetcbas 16775 Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)       (𝜑𝑈 = (Base‘𝐶))

Theoremsetchomfval 16776* Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦𝑚 𝑥)))

Theoremsetchom 16777 Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋𝐻𝑌) = (𝑌𝑚 𝑋))

Theoremelsetchom 16778 A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝑋𝑌))

Theoremsetccofval 16779* Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))

Theoremsetcco 16780 Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑈)    &   (𝜑𝐹:𝑋𝑌)    &   (𝜑𝐺:𝑌𝑍)       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Theoremsetccatid 16781* Lemma for setccat 16782. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝑈 ↦ ( I ↾ 𝑥))))

Theoremsetccat 16782 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremsetcid 16783 The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &    1 = (Id‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)       (𝜑 → ( 1𝑋) = ( I ↾ 𝑋))

Theoremsetcmon 16784 A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝑀 = (Mono‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋1-1𝑌))

Theoremsetcepi 16785 An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝐸 = (Epi‘𝐶)    &   (𝜑 → 2𝑜𝑈)       (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋onto𝑌))

Theoremsetcsect 16786 A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝑆 = (Sect‘𝐶)       (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹:𝑋𝑌𝐺:𝑌𝑋 ∧ (𝐺𝐹) = ( I ↾ 𝑋))))

Theoremsetcinv 16787 An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝑁 = (Inv‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹:𝑋1-1-onto𝑌𝐺 = 𝐹)))

Theoremsetciso 16788 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋1-1-onto𝑌))

Theoremresssetc 16789 The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the SetCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   𝐷 = (SetCat‘𝑉)    &   (𝜑𝑈𝑊)    &   (𝜑𝑉𝑈)       (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))

Theoremfuncsetcres2 16790 A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝐶 = (SetCat‘𝑈)    &   𝐷 = (SetCat‘𝑉)    &   (𝜑𝑈𝑊)    &   (𝜑𝑉𝑈)       (𝜑 → (𝐸 Func 𝐷) ⊆ (𝐸 Func 𝐶))

8.3.2  The category of categories

Syntaxccatc 16791 Extend class notation to include the category Cat.
class CatCat

Definitiondf-catc 16792* Definition of the category Cat, which consists of all categories in the universe 𝑢 (i.e. "𝑢-small categories", see definition 3.44. of [Adamek] p. 39), with functors as the morphisms. Definition 3.47 of [Adamek] p. 40. We do not introduce a specific definition for "𝑢 -large categories", which can be expressed as (Cat ∖ 𝑢). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat = (𝑢 ∈ V ↦ (𝑢 ∩ Cat) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓)))⟩})

Theoremcatcval 16793* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (𝑈 ∩ Cat))    &   (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 Func 𝑦)))    &   (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremcatcbas 16794 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑𝐵 = (𝑈 ∩ Cat))

Theoremcatchomfval 16795* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 Func 𝑦)))

Theoremcatchom 16796 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋 Func 𝑌))

Theoremcatccofval 16797* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔func 𝑓))))

Theoremcatcco 16798 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋 Func 𝑌))    &   (𝜑𝐺 ∈ (𝑌 Func 𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺func 𝐹))

Theoremcatccatid 16799* Lemma for catccat 16801. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ (idfunc𝑥))))

Theoremcatcid 16800 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝐼 = (idfunc𝑋)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)       (𝜑 → ( 1𝑋) = 𝐼)

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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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