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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | mreacs 17701 | Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
| ⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)) | ||
| Theorem | acsfn 17702* | Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
| ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} ∈ (ACS‘𝑋)) | ||
| Theorem | acsfn0 17703* | Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ 𝐾 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
| Theorem | acsfn1 17704* | Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
| Theorem | acsfn1c 17705* | Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝐾 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
| Theorem | acsfn2 17706* | Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) | ||
| Syntax | ccat 17707 | Extend class notation with the class of categories. |
| class Cat | ||
| Syntax | ccid 17708 | Extend class notation with the identity arrow of a category. |
| class Id | ||
| Syntax | chomf 17709 | Extend class notation to include functionalized Hom-set extractor. |
| class Homf | ||
| Syntax | ccomf 17710 | Extend class notation to include functionalized composition operation. |
| class compf | ||
| Definition | df-cat 17711* | A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated with those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53, without the axiom CAT 1, i.e., pairwise disjointness of hom-sets (cat1 18142). See setc2obas 18139 and setc2ohom 18140 for a counterexample. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ((Base‘𝑐)), the morphisms "hom" ((Hom ‘𝑐)) and the composition law "o" ((comp‘𝑐)). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 17712. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.) |
| ⊢ Cat = {𝑐 ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))))} | ||
| Definition | df-cid 17712* | Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ Id = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓)))) | ||
| Definition | df-homf 17713* | Define the functionalized Hom-set operator, which is exactly like Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) | ||
| Definition | df-comf 17714* | Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)))) | ||
| Theorem | iscat 17715* | The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓)))))) | ||
| Theorem | iscatd 17716* | Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | catidex 17717* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) | ||
| Theorem | catideu 17718* | Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓)) | ||
| Theorem | cidfval 17719* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 1 = (𝑥 ∈ 𝐵 ↦ (℩𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(〈𝑥, 𝑥〉 · 𝑦)𝑔) = 𝑓)))) | ||
| Theorem | cidval 17720* | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = (℩𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(〈𝑦, 𝑋〉 · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(〈𝑋, 𝑋〉 · 𝑦)𝑔) = 𝑓))) | ||
| Theorem | cidffn 17721 | The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.) |
| ⊢ Id Fn Cat | ||
| Theorem | cidfn 17722 | The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → 1 Fn 𝐵) | ||
| Theorem | catidd 17723* | Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (〈𝑦, 𝑥〉 · 𝑥)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(〈𝑥, 𝑥〉 · 𝑦) 1 ) = 𝑓) ⇒ ⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ 1 )) | ||
| Theorem | iscatd2 17724* | Version of iscatd 17716 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦)) & ⊢ ((𝜑 ∧ 𝜓) → ( 1 (〈𝑥, 𝑦〉 · 𝑦)𝑓) = 𝑓) & ⊢ ((𝜑 ∧ 𝜓) → (𝑔(〈𝑦, 𝑦〉 · 𝑧) 1 ) = 𝑔) & ⊢ ((𝜑 ∧ 𝜓) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) & ⊢ ((𝜑 ∧ 𝜓) → ((𝑘(〈𝑦, 𝑧〉 · 𝑤)𝑔)(〈𝑥, 𝑦〉 · 𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉 · 𝑤)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓))) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 ))) | ||
| Theorem | catidcl 17725 | Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) | ||
| Theorem | catlid 17726 | Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (( 1 ‘𝑌)(〈𝑋, 𝑌〉 · 𝑌)𝐹) = 𝐹) | ||
| Theorem | catrid 17727 | Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (𝐹(〈𝑋, 𝑋〉 · 𝑌)( 1 ‘𝑋)) = 𝐹) | ||
| Theorem | catcocl 17728 | Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍)) | ||
| Theorem | catass 17729 | Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊)) ⇒ ⊢ (𝜑 → ((𝐾(〈𝑌, 𝑍〉 · 𝑊)𝐺)(〈𝑋, 𝑌〉 · 𝑊)𝐹) = (𝐾(〈𝑋, 𝑍〉 · 𝑊)(𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹))) | ||
| Theorem | catcone0 17730 | Composition of non-empty hom-sets is non-empty. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → (𝑋𝐻𝑌) ≠ ∅) & ⊢ (𝜑 → (𝑌𝐻𝑍) ≠ ∅) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑍) ≠ ∅) | ||
| Theorem | 0catg 17731 | Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat) | ||
| Theorem | 0cat 17732 | The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ ∅ ∈ Cat | ||
| Theorem | homffval 17733* | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
| ⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) | ||
| Theorem | fnhomeqhomf 17734 | If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.) |
| ⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻) | ||
| Theorem | homfval 17735 | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌)) | ||
| Theorem | homffn 17736 | The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝐹 = (Homf ‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐹 Fn (𝐵 × 𝐵) | ||
| Theorem | homfeq 17737* | Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.) |
| ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) ⇒ ⊢ (𝜑 → ((Homf ‘𝐶) = (Homf ‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) = (𝑥𝐽𝑦))) | ||
| Theorem | homfeqd 17738 | If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) & ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) | ||
| Theorem | homfeqbas 17739 | Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷)) | ||
| Theorem | homfeqval 17740 | Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐽𝑌)) | ||
| Theorem | comfffval 17741* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
| ⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) | ||
| Theorem | comffval 17742* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) | ||
| Theorem | comfval 17743 | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) | ||
| Theorem | comfffval2 17744* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) | ||
| Theorem | comffval2 17745* | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) | ||
| Theorem | comfval2 17746 | Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) | ||
| Theorem | comfffn 17747 | The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵) | ||
| Theorem | comffn 17748 | The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝑂 = (compf‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))) | ||
| Theorem | comfeq 17749* | Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → ((compf‘𝐶) = (compf‘𝐷) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) = (𝑔(〈𝑥, 𝑦〉 ∙ 𝑧)𝑓))) | ||
| Theorem | comfeqd 17750 | Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐷)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) | ||
| Theorem | comfeqval 17751 | Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 ∙ 𝑍)𝐹)) | ||
| Theorem | catpropd 17752 | Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat)) | ||
| Theorem | cidpropd 17753 | Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷)) | ||
| Syntax | coppc 17754 | The opposite category operation. |
| class oppCat | ||
| Definition | df-oppc 17755* | Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. Definition 3.5 of [Adamek] p. 25. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet 〈(Hom ‘ndx), tpos (Hom ‘𝑓)〉) sSet 〈(comp‘ndx), (𝑢 ∈ ((Base‘𝑓) × (Base‘𝑓)), 𝑧 ∈ (Base‘𝑓) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝑓)(1st ‘𝑢)))〉)) | ||
| Theorem | oppcval 17756* | Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → 𝑂 = ((𝐶 sSet 〈(Hom ‘ndx), tpos 𝐻〉) sSet 〈(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (〈𝑧, (2nd ‘𝑢)〉 · (1st ‘𝑢)))〉)) | ||
| Theorem | oppchomfval 17757 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.) |
| ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ tpos 𝐻 = (Hom ‘𝑂) | ||
| Theorem | oppchom 17758 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝑋(Hom ‘𝑂)𝑌) = (𝑌𝐻𝑋) | ||
| Theorem | oppccofval 17759 | Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉(comp‘𝑂)𝑍) = tpos (〈𝑍, 𝑌〉 · 𝑋)) | ||
| Theorem | oppcco 17760 | Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(comp‘𝑂)𝑍)𝐹) = (𝐹(〈𝑍, 𝑌〉 · 𝑋)𝐺)) | ||
| Theorem | oppcbas 17761 | Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
| Theorem | oppccatid 17762 | Lemma for oppccat 17765. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → (𝑂 ∈ Cat ∧ (Id‘𝑂) = (Id‘𝐶))) | ||
| Theorem | oppchomf 17763 | Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐶) ⇒ ⊢ tpos 𝐻 = (Homf ‘𝑂) | ||
| Theorem | oppcid 17764 | Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Id‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = 𝐵) | ||
| Theorem | oppccat 17765 | An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) | ||
| Theorem | 2oppcbas 17766 | The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 17781. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ 𝐵 = (Base‘(oppCat‘𝑂)) | ||
| Theorem | 2oppchomf 17767 | The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 17781. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)) | ||
| Theorem | 2oppccomf 17768 | The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 17781. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂)) | ||
| Theorem | oppchomfpropd 17769 | If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) ⇒ ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷))) | ||
| Theorem | oppccomfpropd 17770 | If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷))) | ||
| Theorem | oppccatf 17771 | oppCat restricted to Cat is a function from Cat to Cat. (Contributed by Zhi Wang, 29-Aug-2024.) |
| ⊢ (oppCat ↾ Cat):Cat⟶Cat | ||
| Syntax | cmon 17772 | Extend class notation with the class of all monomorphisms. |
| class Mono | ||
| Syntax | cepi 17773 | Extend class notation with the class of all epimorphisms. |
| class Epi | ||
| Definition | df-mon 17774* | Function returning the monomorphisms of the category 𝑐. JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.) |
| ⊢ Mono = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))})) | ||
| Definition | df-epi 17775 | Function returning the epimorphisms of the category 𝑐. JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.) |
| ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐))) | ||
| Theorem | monfval 17776* | Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(〈𝑧, 𝑥〉 · 𝑦)𝑔))})) | ||
| Theorem | ismon 17777* | Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(〈𝑧, 𝑋〉 · 𝑌)𝑔))))) | ||
| Theorem | ismon2 17778* | Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(〈𝑧, 𝑋〉 · 𝑌)𝑔) = (𝐹(〈𝑧, 𝑋〉 · 𝑌)ℎ) → 𝑔 = ℎ)))) | ||
| Theorem | monhom 17779 | A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) ⊆ (𝑋𝐻𝑌)) | ||
| Theorem | moni 17780 | Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑀 = (Mono‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋)) ⇒ ⊢ (𝜑 → ((𝐹(〈𝑍, 𝑋〉 · 𝑌)𝐺) = (𝐹(〈𝑍, 𝑋〉 · 𝑌)𝐾) ↔ 𝐺 = 𝐾)) | ||
| Theorem | monpropd 17781 | If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘𝐷)) | ||
| Theorem | oppcmon 17782 | A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝑀 = (Mono‘𝑂) & ⊢ 𝐸 = (Epi‘𝐶) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋)) | ||
| Theorem | oppcepi 17783 | An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐸 = (Epi‘𝑂) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋)) | ||
| Theorem | isepi 17784* | Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑌𝐻𝑧) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹))))) | ||
| Theorem | isepi2 17785* | Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐸𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑌𝐻𝑧)∀ℎ ∈ (𝑌𝐻𝑧)((𝑔(〈𝑋, 𝑌〉 · 𝑧)𝐹) = (ℎ(〈𝑋, 𝑌〉 · 𝑧)𝐹) → 𝑔 = ℎ)))) | ||
| Theorem | epihom 17786 | An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) ⊆ (𝑋𝐻𝑌)) | ||
| Theorem | epii 17787 | Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐸 = (Epi‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐸𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → 𝐾 ∈ (𝑌𝐻𝑍)) ⇒ ⊢ (𝜑 → ((𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐾(〈𝑋, 𝑌〉 · 𝑍)𝐹) ↔ 𝐺 = 𝐾)) | ||
| Syntax | csect 17788 | Extend class notation with the sections of a morphism. |
| class Sect | ||
| Syntax | cinv 17789 | Extend class notation with the inverses of a morphism. |
| class Inv | ||
| Syntax | ciso 17790 | Extend class notation with the class of all isomorphisms. |
| class Iso | ||
| Definition | df-sect 17791* | Function returning the section relation in a category. Given arrows 𝑓:𝑋⟶𝑌 and 𝑔:𝑌⟶𝑋, we say 𝑓Sect𝑔, that is, 𝑓 is a section of 𝑔, if 𝑔 ∘ 𝑓 = 1‘𝑋. If there there is an arrow 𝑔 with 𝑓Sect𝑔, the arrow 𝑓 is called a section, see definition 7.19 of [Adamek] p. 106. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {〈𝑓, 𝑔〉 ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))})) | ||
| Definition | df-inv 17792* | The inverse relation in a category. Given arrows 𝑓:𝑋⟶𝑌 and 𝑔:𝑌⟶𝑋, we say 𝑔Inv𝑓, that is, 𝑔 is an inverse of 𝑓, if 𝑔 is a section of 𝑓 and 𝑓 is a section of 𝑔. Definition 3.8 of [Adamek] p. 28. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.) |
| ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) | ||
| Definition | df-iso 17793* | Function returning the isomorphisms of the category 𝑐. Definition 3.8 of [Adamek] p. 28, and definition in [Lang] p. 54. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 2-Jan-2017.) |
| ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) | ||
| Theorem | sectffval 17794* | Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉 · 𝑥)𝑓) = ( 1 ‘𝑥))})) | ||
| Theorem | sectfval 17795* | Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑆𝑌) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}) | ||
| Theorem | sectss 17796 | The section relation is a relation between morphisms from 𝑋 to 𝑌 and morphisms from 𝑌 to 𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) | ||
| Theorem | issect 17797 | The property "𝐹 is a section of 𝐺". (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋)))) | ||
| Theorem | issect2 17798 | Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋)) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐺(〈𝑋, 𝑌〉 · 𝑋)𝐹) = ( 1 ‘𝑋))) | ||
| Theorem | sectcan 17799 | If 𝐺 is a section of 𝐹 and 𝐹 is a section of 𝐻, then 𝐺 = 𝐻. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐺(𝑋𝑆𝑌)𝐹) & ⊢ (𝜑 → 𝐹(𝑌𝑆𝑋)𝐻) ⇒ ⊢ (𝜑 → 𝐺 = 𝐻) | ||
| Theorem | sectco 17800 | Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) & ⊢ (𝜑 → 𝐻(𝑌𝑆𝑍)𝐾) ⇒ ⊢ (𝜑 → (𝐻(〈𝑋, 𝑌〉 · 𝑍)𝐹)(𝑋𝑆𝑍)(𝐺(〈𝑍, 𝑌〉 · 𝑋)𝐾)) | ||
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