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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | funcsetcestrclem3 18201* | Lemma 3 for funcsetcestrc 18209. (Contributed by AV, 27-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) ⇒ ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) | ||
| Theorem | embedsetcestrclem 18202* | Lemma for embedsetcestrc 18212. (Contributed by AV, 31-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) ⇒ ⊢ (𝜑 → 𝐹:𝐶–1-1→𝐵) | ||
| Theorem | funcsetcestrclem4 18203* | Lemma 4 for funcsetcestrc 18209. (Contributed by AV, 27-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶)) | ||
| Theorem | funcsetcestrclem5 18204* | Lemma 5 for funcsetcestrc 18209. (Contributed by AV, 27-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑m 𝑋))) | ||
| Theorem | funcsetcestrclem6 18205* | Lemma 6 for funcsetcestrc 18209. (Contributed by AV, 27-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻) | ||
| Theorem | funcsetcestrclem7 18206* | Lemma 7 for funcsetcestrc 18209. (Contributed by AV, 27-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) & ⊢ 𝐸 = (ExtStrCat‘𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑋𝐺𝑋)‘((Id‘𝑆)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋))) | ||
| Theorem | funcsetcestrclem8 18207* | Lemma 8 for funcsetcestrc 18209. (Contributed by AV, 28-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) & ⊢ 𝐸 = (ExtStrCat‘𝑈) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑆)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐸)(𝐹‘𝑌))) | ||
| Theorem | funcsetcestrclem9 18208* | Lemma 9 for funcsetcestrc 18209. (Contributed by AV, 28-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) & ⊢ 𝐸 = (ExtStrCat‘𝑈) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶 ∧ 𝑍 ∈ 𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(〈𝑋, 𝑌〉(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉(comp‘𝐸)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))) | ||
| Theorem | funcsetcestrc 18209* | The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 28-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) & ⊢ 𝐸 = (ExtStrCat‘𝑈) ⇒ ⊢ (𝜑 → 𝐹(𝑆 Func 𝐸)𝐺) | ||
| Theorem | fthsetcestrc 18210* | The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is faithful. (Contributed by AV, 31-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) & ⊢ 𝐸 = (ExtStrCat‘𝑈) ⇒ ⊢ (𝜑 → 𝐹(𝑆 Faith 𝐸)𝐺) | ||
| Theorem | fullsetcestrc 18211* | The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is full. (Contributed by AV, 1-Apr-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) & ⊢ 𝐸 = (ExtStrCat‘𝑈) ⇒ ⊢ (𝜑 → 𝐹(𝑆 Full 𝐸)𝐺) | ||
| Theorem | embedsetcestrc 18212* | The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is an embedding. According to definition 3.27 (1) of [Adamek] p. 34, a functor "F is called an embedding provided that F is injective on morphisms", or according to remark 3.28 (1) in [Adamek] p. 34, "a functor is an embedding if and only if it is faithful and injective on objects". (Contributed by AV, 31-Mar-2020.) |
| ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑m 𝑥)))) & ⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) ⇒ ⊢ (𝜑 → (𝐹(𝑆 Faith 𝐸)𝐺 ∧ 𝐹:𝐶–1-1→𝐵)) | ||
| Syntax | cxpc 18213 | Extend class notation with the product of two categories. |
| class ×c | ||
| Syntax | c1stf 18214 | Extend class notation with the first projection functor. |
| class 1stF | ||
| Syntax | c2ndf 18215 | Extend class notation with the second projection functor. |
| class 2ndF | ||
| Syntax | cprf 18216 | Extend class notation with the functor pairing operation. |
| class 〈,〉F | ||
| Definition | df-xpc 18217* | Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.) |
| ⊢ ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))〉(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))〉(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))〉))〉}) | ||
| Definition | df-1stf 18218* | Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 1stF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌〈(1st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))〉) | ||
| Definition | df-2ndf 18219* | Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌〈(2nd ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))〉) | ||
| Definition | df-prf 18220* | Define the pairing operation for functors (which takes two functors 𝐹:𝐶⟶𝐷 and 𝐺:𝐶⟶𝐸 and produces (𝐹 〈,〉F 𝐺):𝐶⟶(𝐷 ×c 𝐸)). (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 〈,〉F = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⦋dom (1st ‘𝑓) / 𝑏⦌〈(𝑥 ∈ 𝑏 ↦ 〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ 〈((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)〉))〉) | ||
| Theorem | fnxpc 18221 | The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.) |
| ⊢ ×c Fn (V × V) | ||
| Theorem | xpcval 18222* | Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝑋 = (Base‘𝐶) & ⊢ 𝑌 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 = (𝑋 × 𝑌)) & ⊢ (𝜑 → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))))) & ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))〉 · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))〉 ∙ (2nd ‘𝑦))(2nd ‘𝑓))〉))) ⇒ ⊢ (𝜑 → 𝑇 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐾〉, 〈(comp‘ndx), 𝑂〉}) | ||
| Theorem | xpcbas 18223 | Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝑋 = (Base‘𝐶) & ⊢ 𝑌 = (Base‘𝐷) ⇒ ⊢ (𝑋 × 𝑌) = (Base‘𝑇) | ||
| Theorem | xpchomfval 18224* | Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 𝐾 = (Hom ‘𝑇) ⇒ ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))) | ||
| Theorem | xpchom 18225 | Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 𝐾 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) | ||
| Theorem | relxpchom 18226 | A hom-set in the binary product of categories is a relation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐾 = (Hom ‘𝑇) ⇒ ⊢ Rel (𝑋𝐾𝑌) | ||
| Theorem | xpccofval 18227* | Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 2-Mar-2024.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐾 = (Hom ‘𝑇) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ 𝑂 = (comp‘𝑇) ⇒ ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))〉 · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))〉 ∙ (2nd ‘𝑦))(2nd ‘𝑓))〉)) | ||
| Theorem | xpcco 18228 | Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐾 = (Hom ‘𝑇) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = 〈((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉 ∙ (2nd ‘𝑍))(2nd ‘𝐹))〉) | ||
| Theorem | xpcco1st 18229 | Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐾 = (Hom ‘𝑇) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍)) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → (1st ‘(𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹)) = ((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘𝐹))) | ||
| Theorem | xpcco2nd 18230 | Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐾 = (Hom ‘𝑇) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍)) & ⊢ · = (comp‘𝐷) ⇒ ⊢ (𝜑 → (2nd ‘(𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹)) = ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉 · (2nd ‘𝑍))(2nd ‘𝐹))) | ||
| Theorem | xpchom2 18231 | Value of the set of morphisms in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝑋 = (Base‘𝐶) & ⊢ 𝑌 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ 𝑋) & ⊢ (𝜑 → 𝑁 ∈ 𝑌) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝑄 ∈ 𝑌) & ⊢ 𝐾 = (Hom ‘𝑇) ⇒ ⊢ (𝜑 → (〈𝑀, 𝑁〉𝐾〈𝑃, 𝑄〉) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) | ||
| Theorem | xpcco2 18232 | Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝑋 = (Base‘𝐶) & ⊢ 𝑌 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ 𝑋) & ⊢ (𝜑 → 𝑁 ∈ 𝑌) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝑄 ∈ 𝑌) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐷) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) & ⊢ (𝜑 → 𝐹 ∈ (𝑀𝐻𝑃)) & ⊢ (𝜑 → 𝐺 ∈ (𝑁𝐽𝑄)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃𝐻𝑅)) & ⊢ (𝜑 → 𝐿 ∈ (𝑄𝐽𝑆)) ⇒ ⊢ (𝜑 → (〈𝐾, 𝐿〉(〈〈𝑀, 𝑁〉, 〈𝑃, 𝑄〉〉𝑂〈𝑅, 𝑆〉)〈𝐹, 𝐺〉) = 〈(𝐾(〈𝑀, 𝑃〉 · 𝑅)𝐹), (𝐿(〈𝑁, 𝑄〉 ∙ 𝑆)𝐺)〉) | ||
| Theorem | xpccatid 18233* | The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑋 = (Base‘𝐶) & ⊢ 𝑌 = (Base‘𝐷) & ⊢ 𝐼 = (Id‘𝐶) & ⊢ 𝐽 = (Id‘𝐷) ⇒ ⊢ (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐼‘𝑥), (𝐽‘𝑦)〉))) | ||
| Theorem | xpcid 18234 | The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑋 = (Base‘𝐶) & ⊢ 𝑌 = (Base‘𝐷) & ⊢ 𝐼 = (Id‘𝐶) & ⊢ 𝐽 = (Id‘𝐷) & ⊢ 1 = (Id‘𝑇) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ∈ 𝑌) ⇒ ⊢ (𝜑 → ( 1 ‘〈𝑅, 𝑆〉) = 〈(𝐼‘𝑅), (𝐽‘𝑆)〉) | ||
| Theorem | xpccat 18235 | The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → 𝑇 ∈ Cat) | ||
| Theorem | 1stfval 18236* | Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐻 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑃 = (𝐶 1stF 𝐷) ⇒ ⊢ (𝜑 → 𝑃 = 〈(1st ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))〉) | ||
| Theorem | 1stf1 18237 | Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐻 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑃 = (𝐶 1stF 𝐷) & ⊢ (𝜑 → 𝑅 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((1st ‘𝑃)‘𝑅) = (1st ‘𝑅)) | ||
| Theorem | 1stf2 18238 | Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐻 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑃 = (𝐶 1stF 𝐷) & ⊢ (𝜑 → 𝑅 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑅(2nd ‘𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆))) | ||
| Theorem | 2ndfval 18239* | Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐻 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑄 = (𝐶 2ndF 𝐷) ⇒ ⊢ (𝜑 → 𝑄 = 〈(2nd ↾ 𝐵), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))〉) | ||
| Theorem | 2ndf1 18240 | Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐻 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑄 = (𝐶 2ndF 𝐷) & ⊢ (𝜑 → 𝑅 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((1st ‘𝑄)‘𝑅) = (2nd ‘𝑅)) | ||
| Theorem | 2ndf2 18241 | Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ 𝐻 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑄 = (𝐶 2ndF 𝐷) & ⊢ (𝜑 → 𝑅 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑅(2nd ‘𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆))) | ||
| Theorem | 1stfcl 18242 | The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑃 = (𝐶 1stF 𝐷) ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝑇 Func 𝐶)) | ||
| Theorem | 2ndfcl 18243 | The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑄 = (𝐶 2ndF 𝐷) ⇒ ⊢ (𝜑 → 𝑄 ∈ (𝑇 Func 𝐷)) | ||
| Theorem | prfval 18244* | Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) ⇒ ⊢ (𝜑 → 𝑃 = 〈(𝑥 ∈ 𝐵 ↦ 〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (ℎ ∈ (𝑥𝐻𝑦) ↦ 〈((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)〉))〉) | ||
| Theorem | prf1 18245 | Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((1st ‘𝑃)‘𝑋) = 〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐺)‘𝑋)〉) | ||
| Theorem | prf2fval 18246* | Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋(2nd ‘𝑃)𝑌) = (ℎ ∈ (𝑋𝐻𝑌) ↦ 〈((𝑋(2nd ‘𝐹)𝑌)‘ℎ), ((𝑋(2nd ‘𝐺)𝑌)‘ℎ)〉)) | ||
| Theorem | prf2 18247 | Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → ((𝑋(2nd ‘𝑃)𝑌)‘𝐾) = 〈((𝑋(2nd ‘𝐹)𝑌)‘𝐾), ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)〉) | ||
| Theorem | prfcl 18248 | The pairing of functors 𝐹:𝐶⟶𝐷 and 𝐺:𝐶⟶𝐷 is a functor 〈𝐹, 𝐺〉:𝐶⟶(𝐷 × 𝐸). (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) & ⊢ 𝑇 = (𝐷 ×c 𝐸) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) ⇒ ⊢ (𝜑 → 𝑃 ∈ (𝐶 Func 𝑇)) | ||
| Theorem | prf1st 18249 | Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) ⇒ ⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹) | ||
| Theorem | prf2nd 18250 | Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝑃 = (𝐹 〈,〉F 𝐺) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸)) ⇒ ⊢ (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝑃) = 𝐺) | ||
| Theorem | 1st2ndprf 18251 | Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝑇 = (𝐷 ×c 𝐸) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝑇)) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐸 ∈ Cat) ⇒ ⊢ (𝜑 → 𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) 〈,〉F ((𝐷 2ndF 𝐸) ∘func 𝐹))) | ||
| Theorem | catcxpccl 18252 | The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.) |
| ⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑇 = (𝑋 ×c 𝑌) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑇 ∈ 𝐵) | ||
| Theorem | xpcpropd 18253 | If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.) |
| ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) & ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷)) | ||
| Syntax | cevlf 18254 | Extend class notation with the evaluation functor. |
| class evalF | ||
| Syntax | ccurf 18255 | Extend class notation with the currying of a functor. |
| class curryF | ||
| Syntax | cuncf 18256 | Extend class notation with the uncurrying of a functor. |
| class uncurryF | ||
| Syntax | cdiag 18257 | Extend class notation to include the diagonal functor. |
| class Δfunc | ||
| Definition | df-evlf 18258* | Define the evaluation functor, which is the extension of the evaluation map 𝑓, 𝑥 ↦ (𝑓‘𝑥) of functors, to a functor (𝐶⟶𝐷) × 𝐶⟶𝐷. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ evalF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ 〈(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(〈((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))〉(comp‘𝑑)((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))〉) | ||
| Definition | df-curf 18259* | Define the curry functor, which maps a functor 𝐹:𝐶 × 𝐷⟶𝐸 to curryF (𝐹):𝐶⟶(𝐷⟶𝐸). (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑒) / 𝑐⦌⦋(2nd ‘𝑒) / 𝑑⦌〈(𝑥 ∈ (Base‘𝑐) ↦ 〈(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st ‘𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝑓)〈𝑥, 𝑧〉)𝑔)))〉), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(〈𝑥, 𝑧〉(2nd ‘𝑓)〈𝑦, 𝑧〉)((Id‘𝑑)‘𝑧)))))〉) | ||
| Definition | df-uncf 18260* | Define the uncurry functor, which can be defined equationally using evalF. Strictly speaking, the third category argument is not needed, since the resulting functor is extensionally equal regardless, but it is used in the equational definition and is too much work to remove. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) 〈,〉F ((𝑐‘0) 2ndF (𝑐‘1))))) | ||
| Definition | df-diag 18261* | Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑥). The value of the functor at an object 𝑥 is the constant functor which maps all objects in 𝐷 to 𝑥 and all morphisms to 1(𝑥). The morphism part is a natural transformation between these functors, which takes 𝑓:𝑥⟶𝑦 to the natural transformation with every component equal to 𝑓. (Contributed by Mario Carneiro, 6-Jan-2017.) |
| ⊢ Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (〈𝑐, 𝑑〉 curryF (𝑐 1stF 𝑑))) | ||
| Theorem | evlfval 18262* | Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐸 = (𝐶 evalF 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) ⇒ ⊢ (𝜑 → 𝐸 = 〈(𝑓 ∈ (𝐶 Func 𝐷), 𝑥 ∈ 𝐵 ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ ⦋(1st ‘𝑥) / 𝑚⦌⦋(1st ‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(〈((1st ‘𝑚)‘(2nd ‘𝑥)), ((1st ‘𝑚)‘(2nd ‘𝑦))〉 · ((1st ‘𝑛)‘(2nd ‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))〉) | ||
| Theorem | evlf2 18263* | Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐸 = (𝐶 evalF 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐿 = (〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑌〉) ⇒ ⊢ (𝜑 → 𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎‘𝑌)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)〉 · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝑔)))) | ||
| Theorem | evlf2val 18264 | Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐸 = (𝐶 evalF 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐿 = (〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑌〉) & ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺)) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)〉 · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾))) | ||
| Theorem | evlf1 18265 | Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐸 = (𝐶 evalF 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹(1st ‘𝐸)𝑋) = ((1st ‘𝐹)‘𝑋)) | ||
| Theorem | evlfcllem 18266 | Lemma for evlfcl 18267. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐸 = (𝐶 evalF 𝐷) & ⊢ 𝑄 = (𝐶 FuncCat 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (Base‘𝐶))) & ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (Base‘𝐶))) & ⊢ (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝑍 ∈ (Base‘𝐶))) & ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌))) & ⊢ (𝜑 → (𝐵 ∈ (𝐺𝑁𝐻) ∧ 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍))) ⇒ ⊢ (𝜑 → ((〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐻, 𝑍〉)‘(〈𝐵, 𝐿〉(〈〈𝐹, 𝑋〉, 〈𝐺, 𝑌〉〉(comp‘(𝑄 ×c 𝐶))〈𝐻, 𝑍〉)〈𝐴, 𝐾〉)) = (((〈𝐺, 𝑌〉(2nd ‘𝐸)〈𝐻, 𝑍〉)‘〈𝐵, 𝐿〉)(〈((1st ‘𝐸)‘〈𝐹, 𝑋〉), ((1st ‘𝐸)‘〈𝐺, 𝑌〉)〉(comp‘𝐷)((1st ‘𝐸)‘〈𝐻, 𝑍〉))((〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑌〉)‘〈𝐴, 𝐾〉))) | ||
| Theorem | evlfcl 18267 | The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors 𝐶⟶𝐷, and the second parameter in 𝐷. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐸 = (𝐶 evalF 𝐷) & ⊢ 𝑄 = (𝐶 FuncCat 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷)) | ||
| Theorem | curfval 18268* | Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) ⇒ ⊢ (𝜑 → 𝐺 = 〈(𝑥 ∈ 𝐴 ↦ 〈(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑧〉)𝑔)))〉), (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧 ∈ 𝐵 ↦ (𝑔(〈𝑥, 𝑧〉(2nd ‘𝐹)〈𝑦, 𝑧〉)(𝐼‘𝑧)))))〉) | ||
| Theorem | curf1fval 18269* | Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → (1st ‘𝐺) = (𝑥 ∈ 𝐴 ↦ 〈(𝑦 ∈ 𝐵 ↦ (𝑥(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝐹)〈𝑥, 𝑧〉)𝑔)))〉)) | ||
| Theorem | curf1 18270* | Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐾 = 〈(𝑦 ∈ 𝐵 ↦ (𝑋(1st ‘𝐹)𝑦)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1 ‘𝑋)(〈𝑋, 𝑦〉(2nd ‘𝐹)〈𝑋, 𝑧〉)𝑔)))〉) | ||
| Theorem | curf11 18271 | Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑋(1st ‘𝐹)𝑌)) | ||
| Theorem | curf12 18272 | The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍)) ⇒ ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (( 1 ‘𝑋)(〈𝑋, 𝑌〉(2nd ‘𝐹)〈𝑋, 𝑍〉)𝐻)) | ||
| Theorem | curf1cl 18273 | The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐺)‘𝑋) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸)) | ||
| Theorem | curf2 18274* | Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) ⇒ ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ (𝐾(〈𝑋, 𝑧〉(2nd ‘𝐹)〈𝑌, 𝑧〉)(𝐼‘𝑧)))) | ||
| Theorem | curf2val 18275 | Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐿‘𝑍) = (𝐾(〈𝑋, 𝑍〉(2nd ‘𝐹)〈𝑌, 𝑍〉)(𝐼‘𝑍))) | ||
| Theorem | curf2cl 18276 | The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾) & ⊢ 𝑁 = (𝐷 Nat 𝐸) ⇒ ⊢ (𝜑 → 𝐿 ∈ (((1st ‘𝐺)‘𝑋)𝑁((1st ‘𝐺)‘𝑌))) | ||
| Theorem | curfcl 18277 | The curry functor of a functor 𝐹:𝐶 × 𝐷⟶𝐸 is a functor curryF (𝐹):𝐶⟶(𝐷⟶𝐸). (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ 𝑄 = (𝐷 FuncCat 𝐸) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝑄)) | ||
| Theorem | curfpropd 18278 | If two categories have the same set of objects, morphisms, and compositions, then they curry the same functor to the same result. (Contributed by Mario Carneiro, 26-Jan-2017.) |
| ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) & ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐴 ∈ Cat) & ⊢ (𝜑 → 𝐵 ∈ Cat) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸)) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐶〉 curryF 𝐹) = (〈𝐵, 𝐷〉 curryF 𝐹)) | ||
| Theorem | uncfval 18279 | Value of the uncurry functor, which is the reverse of the curry functor, taking 𝐺:𝐶⟶(𝐷⟶𝐸) to uncurryF (𝐺):𝐶 × 𝐷⟶𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐹 = (〈“𝐶𝐷𝐸”〉 uncurryF 𝐺) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐸 ∈ Cat) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) ⇒ ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) 〈,〉F (𝐶 2ndF 𝐷)))) | ||
| Theorem | uncfcl 18280 | The uncurry operation takes a functor 𝐹:𝐶⟶(𝐷⟶𝐸) to a functor uncurryF (𝐹):𝐶 × 𝐷⟶𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐹 = (〈“𝐶𝐷𝐸”〉 uncurryF 𝐺) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐸 ∈ Cat) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | ||
| Theorem | uncf1 18281 | Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐹 = (〈“𝐶𝐷𝐸”〉 uncurryF 𝐺) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐸 ∈ Cat) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋(1st ‘𝐹)𝑌) = ((1st ‘((1st ‘𝐺)‘𝑋))‘𝑌)) | ||
| Theorem | uncf2 18282 | Value of the uncurry functor on a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐹 = (〈“𝐶𝐷𝐸”〉 uncurryF 𝐺) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐸 ∈ Cat) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑍)) & ⊢ (𝜑 → 𝑆 ∈ (𝑌𝐽𝑊)) ⇒ ⊢ (𝜑 → (𝑅(〈𝑋, 𝑌〉(2nd ‘𝐹)〈𝑍, 𝑊〉)𝑆) = ((((𝑋(2nd ‘𝐺)𝑍)‘𝑅)‘𝑊)(〈((1st ‘((1st ‘𝐺)‘𝑋))‘𝑌), ((1st ‘((1st ‘𝐺)‘𝑋))‘𝑊)〉(comp‘𝐸)((1st ‘((1st ‘𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st ‘𝐺)‘𝑋))𝑊)‘𝑆))) | ||
| Theorem | curfuncf 18283 | Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐹 = (〈“𝐶𝐷𝐸”〉 uncurryF 𝐺) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐸 ∈ Cat) & ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) ⇒ ⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF 𝐹) = 𝐺) | ||
| Theorem | uncfcurf 18284 | Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.) |
| ⊢ 𝐺 = (〈𝐶, 𝐷〉 curryF 𝐹) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) ⇒ ⊢ (𝜑 → (〈“𝐶𝐷𝐸”〉 uncurryF 𝐺) = 𝐹) | ||
| Theorem | diagval 18285 | Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑥). We can define this equationally as the currying of the first projection functor, and by expressing it this way we get a quick proof of functoriality. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → 𝐿 = (〈𝐶, 𝐷〉 curryF (𝐶 1stF 𝐷))) | ||
| Theorem | diagcl 18286 | The diagonal functor is a functor from the base category to the functor category. Another way of saying this is that the constant functor (𝑦 ∈ 𝐷 ↦ 𝑋) is a construction that is natural in 𝑋 (and covariant). (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑄 = (𝐷 FuncCat 𝐶) ⇒ ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func 𝑄)) | ||
| Theorem | diag1cl 18287 | The constant functor of 𝑋 is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶)) | ||
| Theorem | diag11 18288 | Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋) | ||
| Theorem | diag12 18289 | Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑌𝐽𝑍)) ⇒ ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐹) = ( 1 ‘𝑋)) | ||
| Theorem | diag2 18290 | Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹})) | ||
| Theorem | diag2cl 18291 | The diagonal functor at a morphism is a natural transformation between constant functors. (Contributed by Mario Carneiro, 7-Jan-2017.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ 𝑁 = (𝐷 Nat 𝐶) ⇒ ⊢ (𝜑 → (𝐵 × {𝐹}) ∈ (((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌))) | ||
| Theorem | curf2ndf 18292 | As shown in diagval 18285, the currying of the first projection is the diagonal functor. On the other hand, the currying of the second projection is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑦), which is a constant functor of the identity functor at 𝐷. (Contributed by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝑄 = (𝐷 FuncCat 𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (〈𝐶, 𝐷〉 curryF (𝐶 2ndF 𝐷)) = ((1st ‘(𝑄Δfunc𝐶))‘(idfunc‘𝐷))) | ||
| Syntax | chof 18293 | Extend class notation with the Hom functor. |
| class HomF | ||
| Syntax | cyon 18294 | Extend class notation with the Yoneda embedding. |
| class Yon | ||
| Definition | df-hof 18295* | Define the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from (oppCat‘𝐶) × 𝐶 to SetCat, whose object part is the hom-function Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ HomF = (𝑐 ∈ Cat ↦ 〈(Homf ‘𝑐), ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝑐)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd ‘𝑦))ℎ)(〈(1st ‘𝑦), (1st ‘𝑥)〉(comp‘𝑐)(2nd ‘𝑦))𝑓))))〉) | ||
| Definition | df-yon 18296 | Define the Yoneda embedding, which is the currying of the (opposite) Hom functor. (Contributed by Mario Carneiro, 11-Jan-2017.) |
| ⊢ Yon = (𝑐 ∈ Cat ↦ (〈𝑐, (oppCat‘𝑐)〉 curryF (HomF‘(oppCat‘𝑐)))) | ||
| Theorem | hofval 18297* | Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from (oppCat‘𝐶) × 𝐶 to SetCat, whose object part is the hom-function Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝑀 = (HomF‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → 𝑀 = 〈(Homf ‘𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st ‘𝑦)𝐻(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(〈(1st ‘𝑦), (1st ‘𝑥)〉 · (2nd ‘𝑦))𝑓))))〉) | ||
| Theorem | hof1fval 18298 | The object part of the Hom functor is the Homf operation, which is just a functionalized version of Hom. That is, it is a two argument function, which maps 𝑋, 𝑌 to the set of morphisms from 𝑋 to 𝑌. (Contributed by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝑀 = (HomF‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → (1st ‘𝑀) = (Homf ‘𝐶)) | ||
| Theorem | hof1 18299 | The object part of the Hom functor maps 𝑋, 𝑌 to the set of morphisms from 𝑋 to 𝑌. (Contributed by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝑀 = (HomF‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋(1st ‘𝑀)𝑌) = (𝑋𝐻𝑌)) | ||
| Theorem | hof2fval 18300* | The morphism part of the Hom functor, for morphisms 〈𝑓, 𝑔〉:〈𝑋, 𝑌〉⟶〈𝑍, 𝑊〉 (which since the first argument is contravariant means morphisms 𝑓:𝑍⟶𝑋 and 𝑔:𝑌⟶𝑊), yields a function (a morphism of SetCat) mapping ℎ:𝑋⟶𝑌 to 𝑔 ∘ ℎ ∘ 𝑓:𝑍⟶𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.) |
| ⊢ 𝑀 = (HomF‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉(2nd ‘𝑀)〈𝑍, 𝑊〉) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(〈𝑋, 𝑌〉 · 𝑊)ℎ)(〈𝑍, 𝑋〉 · 𝑊)𝑓)))) | ||
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