HomeTheorem List (p. 179 of 469)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | istermo 17801* | 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 17802 | 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 17803* | 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 17804* | 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 17805 | 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 17806 | 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 17807 | An initial object is a terminal object in the opposite category. An alternate definition of df-inito 17788 depending on df-termo 17789. (Contributed by Zhi Wang, 29-Aug-2024.) |
| ⊢ InitO = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐))) | ||
| Theorem | dftermo2 17808 | A terminal object is an initial object in the opposite category. An alternate definition of df-termo 17789 depending on df-inito 17788. (Contributed by Zhi Wang, 29-Aug-2024.) |
| ⊢ TermO = (𝑐 ∈ Cat ↦ (InitO‘(oppCat‘𝑐))) | ||
| Theorem | dfinito3 17809 | An alternate definition of df-inito 17788 depending on df-termo 17789, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.) |
| ⊢ InitO = (TermO ∘ (oppCat ↾ Cat)) | ||
| Theorem | dftermo3 17810 | An alternate definition of df-termo 17789 depending on df-inito 17788, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.) |
| ⊢ TermO = (InitO ∘ (oppCat ↾ Cat)) | ||
| Theorem | initoo 17811 | An initial object is an object. (Contributed by AV, 14-Apr-2020.) |
| ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶))) | ||
| Theorem | termoo 17812 | A terminal object is an object. (Contributed by AV, 18-Apr-2020.) |
| ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶))) | ||
| Theorem | iszeroi 17813 | 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 17814 | 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 17815* | 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 17816 | 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 17817 | Lemma 0 for initoeu2 17820. (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 17818* | Lemma 1 for initoeu2 17820. (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 17819* | Lemma 2 for initoeu2 17820. (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 17820 | 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 17821 | 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 17822* | 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 17823 | 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) | ||
| Syntax | cdoma 17824 | Extend class notation to include the domain extractor for an arrow. |
| class doma | ||
| Syntax | ccoda 17825 | Extend class notation to include the codomain extractor for an arrow. |
| class coda | ||
| Syntax | carw 17826 | Extend class notation to include the collection of all arrows of a category. |
| class Arrow | ||
| Syntax | choma 17827 | Extend class notation to include the set of all arrows with a specific domain and codomain. |
| class Homa | ||
| Definition | df-doma 17828 | 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 17829 | 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 17830* | 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 17828 and df-coda 17829. (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 17831 | 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 17832 | Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐻 = (Homa‘𝐶) ⇒ ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) | ||
| Theorem | homafval 17833* | 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 17834 | 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 17835 | 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 17836 | 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 17837 | 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 17838 | 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 17839 | 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 17840 | An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐻 = (Homa‘𝐶) ⇒ ⊢ Rel (𝑋𝐻𝑌) | ||
| Theorem | homa1 17841 | 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 17842 | 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 17843 | 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 17844 | 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 17845 | 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 17846 | 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 17847 | 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 17848 | 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 17849 | 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 17850 | 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 17851 | 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 17852 | 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 17853 | 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 17854 | 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 17855 | 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 17856 | 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)⟩) | ||
| Syntax | cida 17857 | Extend class notation to include identity for arrows. |
| class Ida | ||
| Syntax | ccoa 17858 | Extend class notation to include composition for arrows. |
| class compa | ||
| Definition | df-ida 17859* | 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)⟩)) | ||
| Definition | df-coa 17860* | 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 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))⟩)) | ||
| Theorem | idafval 17861* | Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩)) | ||
| Theorem | idaval 17862 | Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) = ⟨𝑋, 𝑋, ( 1 ‘𝑋)⟩) | ||
| Theorem | ida2 17863 | Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (2nd ‘(𝐼‘𝑋)) = ( 1 ‘𝑋)) | ||
| Theorem | idahom 17864 | Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐻 = (Homa‘𝐶) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐻𝑋)) | ||
| Theorem | idadm 17865 | Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (doma‘(𝐼‘𝑋)) = 𝑋) | ||
| Theorem | idacd 17866 | Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (coda‘(𝐼‘𝑋)) = 𝑋) | ||
| Theorem | idaf 17867 | The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐼 = (Ida‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ (𝜑 → 𝐼:𝐵⟶𝐴) | ||
| Theorem | coafval 17868* | The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ · = (compa‘𝐶) & ⊢ 𝐴 = (Arrow‘𝐶) & ⊢ ∙ = (comp‘𝐶) ⇒ ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩) | ||
| Theorem | eldmcoa 17869 | 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.) |
| ⊢ · = (compa‘𝐶) & ⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ (coda‘𝐹) = (doma‘𝐺))) | ||
| Theorem | dmcoass 17870 | The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ · = (compa‘𝐶) & ⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ dom · ⊆ (𝐴 × 𝐴) | ||
| Theorem | homdmcoa 17871 | If 𝐹:𝑋⟶𝑌 and 𝐺:𝑌⟶𝑍, then 𝐺 and 𝐹 are composable. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ · = (compa‘𝐶) & ⊢ 𝐻 = (Homa‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → 𝐺dom · 𝐹) | ||
| Theorem | coaval 17872 | Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ · = (compa‘𝐶) & ⊢ 𝐻 = (Homa‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ ∙ = (comp‘𝐶) ⇒ ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩) | ||
| Theorem | coa2 17873 | The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ · = (compa‘𝐶) & ⊢ 𝐻 = (Homa‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ ∙ = (comp‘𝐶) ⇒ ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))) | ||
| Theorem | coahom 17874 | The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ · = (compa‘𝐶) & ⊢ 𝐻 = (Homa‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍)) | ||
| Theorem | coapm 17875 | Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ · = (compa‘𝐶) & ⊢ 𝐴 = (Arrow‘𝐶) ⇒ ⊢ · ∈ (𝐴 ↑pm (𝐴 × 𝐴)) | ||
| Theorem | arwlid 17876 | Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐻 = (Homa‘𝐶) & ⊢ · = (compa‘𝐶) & ⊢ 1 = (Ida‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹) | ||
| Theorem | arwrid 17877 | Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐻 = (Homa‘𝐶) & ⊢ · = (compa‘𝐶) & ⊢ 1 = (Ida‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 𝐹) | ||
| Theorem | arwass 17878 | Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝐻 = (Homa‘𝐶) & ⊢ · = (compa‘𝐶) & ⊢ 1 = (Ida‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊)) ⇒ ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹))) | ||
| Syntax | csetc 17879 | Extend class notation to include the category Set. |
| class SetCat | ||
| Definition | df-setc 17880* | 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), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩}) | ||
| Theorem | setcval 17881* | Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑m 𝑥))) & ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) ⇒ ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}) | ||
| Theorem | setcbas 17882 | Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑈 = (Base‘𝐶)) | ||
| Theorem | setchomfval 17883* | Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑m 𝑥))) | ||
| Theorem | setchom 17884 | Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑌 ↑m 𝑋)) | ||
| Theorem | elsetchom 17885 | A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ↔ 𝐹:𝑋⟶𝑌)) | ||
| Theorem | setccofval 17886* | Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) | ||
| Theorem | setcco 17887 | Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑍) ⇒ ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
| Theorem | setccatid 17888* | Lemma for setccat 17889. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥)))) | ||
| Theorem | setccat 17889 | The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
| Theorem | setcid 17890 | 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑋)) | ||
| Theorem | setcmon 17891 | A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹:𝑋–1-1→𝑌)) | ||
| Theorem | setcepi 17892 | An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 2o ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ 𝐹:𝑋–onto→𝑌)) | ||
| Theorem | setcsect 17893 | A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋)))) | ||
| Theorem | setcinv 17894 | An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝑁 = (Inv‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ 𝐺 = ◡𝐹))) | ||
| Theorem | setciso 17895 | An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹:𝑋–1-1-onto→𝑌)) | ||
| Theorem | resssetc 17896 | 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → 𝑉 ⊆ 𝑈) ⇒ ⊢ (𝜑 → ((Homf ‘(𝐶 ↾s 𝑉)) = (Homf ‘𝐷) ∧ (compf‘(𝐶 ↾s 𝑉)) = (compf‘𝐷))) | ||
| Theorem | funcsetcres2 17897 | 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → 𝑉 ⊆ 𝑈) ⇒ ⊢ (𝜑 → (𝐸 Func 𝐷) ⊆ (𝐸 Func 𝐶)) | ||
| Theorem | setc2obas 17898 | ∅ and 1o are distinct objects in (SetCat‘2o). This combined with setc2ohom 17899 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17466 and cat1 17901. (Contributed by Zhi Wang, 24-Sep-2024.) |
| ⊢ 𝐶 = (SetCat‘2o) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅) | ||
| Theorem | setc2ohom 17899 | (SetCat‘2o) is a category (provable from setccat 17889 and 2oex 8370) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 17898. Notably, the empty set ∅ is simultaneously an object (setc2obas 17898) , an identity morphism from ∅ to ∅ (setcid 17890 or thincid 46654) , and a non-identity morphism from ∅ to 1o. See cat1lem 17900 and cat1 17901 for a more general statement. This category is also thin (setc2othin 46677), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 46675 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.) |
| ⊢ 𝐶 = (SetCat‘2o) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o)) | ||
| Theorem | cat1lem 17900* | The category of sets in a "universe" containing the empty set and another set does not have pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. Lemma for cat1 17901. (Contributed by Zhi Wang, 15-Sep-2024.) |
| ⊢ 𝐶 = (SetCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → ∅ ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → ∅ ≠ 𝑌) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | ||
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