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Type | Label | Description |
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Statement | ||
Theorem | funcestrcsetc 17401* | The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 23-Mar-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) ⇒ ⊢ (𝜑 → 𝐹(𝐸 Func 𝑆)𝐺) | ||
Theorem | fthestrcsetc 17402* | The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is faithful. (Contributed by AV, 2-Apr-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) ⇒ ⊢ (𝜑 → 𝐹(𝐸 Faith 𝑆)𝐺) | ||
Theorem | fullestrcsetc 17403* | The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is full. (Contributed by AV, 2-Apr-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) ⇒ ⊢ (𝜑 → 𝐹(𝐸 Full 𝑆)𝐺) | ||
Theorem | equivestrcsetc 17404* | The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is an equivalence. According to definition 3.33 (1) of [Adamek] p. 36, "A functor F : A -> B is called an equivalence provided that it is full, faithful, and isomorphism-dense in the sense that for any B-object B' there exists some A-object A' such that F(A') is isomorphic to B'.". Therefore, the category of sets and the category of extensible structures are equivalent, according to definition 3.33 (2) of [Adamek] p. 36, "Categories A and B are called equivalent provided that there is an equivalence from A to B.". (Contributed by AV, 2-Apr-2020.) |
⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))) & ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐹(𝐸 Faith 𝑆)𝐺 ∧ 𝐹(𝐸 Full 𝑆)𝐺 ∧ ∀𝑏 ∈ 𝐶 ∃𝑎 ∈ 𝐵 ∃𝑖 𝑖:𝑏–1-1-onto→(𝐹‘𝑎))) | ||
Theorem | setc1strwun 17405 | A constructed one-slot structure with the objects of the category of sets as base set in a weak universe. (Contributed by AV, 27-Mar-2020.) |
⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → ω ∈ 𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ 𝑈) | ||
Theorem | funcsetcestrclem1 17406* | Lemma 1 for funcsetcestrc 17416. (Contributed by AV, 27-Mar-2020.) |
⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) | ||
Theorem | funcsetcestrclem2 17407* | Lemma 2 for funcsetcestrc 17416. (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 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) ∈ 𝑈) | ||
Theorem | funcsetcestrclem3 17408* | Lemma 3 for funcsetcestrc 17416. (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 17409* | Lemma for embedsetcestrc 17419. (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 17410* | Lemma 4 for funcsetcestrc 17416. (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 17411* | Lemma 5 for funcsetcestrc 17416. (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 17412* | Lemma 6 for funcsetcestrc 17416. (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 17413* | Lemma 7 for funcsetcestrc 17416. (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 17414* | Lemma 8 for funcsetcestrc 17416. (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 17415* | Lemma 9 for funcsetcestrc 17416. (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 17416* | 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 17417* | 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 17418* | 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 17419* | 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 17420 | Extend class notation with the product of two categories. |
class ×c | ||
Syntax | c1stf 17421 | Extend class notation with the first projection functor. |
class 1stF | ||
Syntax | c2ndf 17422 | Extend class notation with the second projection functor. |
class 2ndF | ||
Syntax | cprf 17423 | Extend class notation with the functor pairing operation. |
class 〈,〉F | ||
Definition | df-xpc 17424* | 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 17425* | 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 17426* | 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 17427* | 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 17428 | The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.) |
⊢ ×c Fn (V × V) | ||
Theorem | xpcval 17429* | 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 17430 | 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 17431* | 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 17432 | 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 17433 | 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 17434* | 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 17435 | 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 17436 | 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 17437 | 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 17438 | 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 17439 | 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 17440* | 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 17441 | 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 17442 | 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 17443* | 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 17444 | 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 17445 | 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 17446* | 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 17447 | 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 17448 | 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 17449 | 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 17450 | 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 17451* | 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 17452 | 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 17453* | 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 17454 | 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 17455 | 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 17456 | 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 17457 | 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 17458 | 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 17459 | The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ 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 17460 | 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 17461 | Extend class notation with the evaluation functor. |
class evalF | ||
Syntax | ccurf 17462 | Extend class notation with the currying of a functor. |
class curryF | ||
Syntax | cuncf 17463 | Extend class notation with the uncurrying of a functor. |
class uncurryF | ||
Syntax | cdiag 17464 | Extend class notation to include the diagonal functor. |
class Δfunc | ||
Definition | df-evlf 17465* | 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 17466* | 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 17467* | 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 17468* | 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 17469* | 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 17470* | 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 17471 | 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 17472 | 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 17473 | Lemma for evlfcl 17474. (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 17474 | 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 17475* | 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 17476* | 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 17477* | 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 17478 | 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 17479 | 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 17480 | 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 17481* | 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 17482 | 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 17483 | 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 17484 | 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 17485 | 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 17486 | 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 17487 | 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 17488 | 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 17489 | 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 17490 | 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 17491 | 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 17492 | 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 17493 | 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 17494 | 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 17495 | 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 17496 | 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 17497 | 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 17498 | 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 17499 | As shown in diagval 17492, 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 17500 | Extend class notation with the Hom functor. |
class HomF |
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