![]() |
Metamath
Proof Explorer Theorem List (p. 183 of 491) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30946) |
![]() (30947-32469) |
![]() (32470-49035) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | funcestrcsetclem7 18201* | Lemma 7 for funcestrcsetc 18204. (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 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝐸)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) | ||
Theorem | funcestrcsetclem8 18202* | Lemma 8 for funcestrcsetc 18204. (Contributed by AV, 15-Feb-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 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌))) | ||
Theorem | funcestrcsetclem9 18203* | Lemma 9 for funcestrcsetc 18204. (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 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝐸)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝐸)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(〈𝑋, 𝑌〉(comp‘𝐸)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))) | ||
Theorem | funcestrcsetc 18204* | 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 18205* | 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 18206* | 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 18207* | 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 18208 | 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 18209* | Lemma 1 for funcsetcestrc 18219. (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 18210* | Lemma 2 for funcsetcestrc 18219. (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 18211* | Lemma 3 for funcsetcestrc 18219. (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 18212* | Lemma for embedsetcestrc 18222. (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 18213* | Lemma 4 for funcsetcestrc 18219. (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 18214* | Lemma 5 for funcsetcestrc 18219. (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 18215* | Lemma 6 for funcsetcestrc 18219. (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 18216* | Lemma 7 for funcsetcestrc 18219. (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 18217* | Lemma 8 for funcsetcestrc 18219. (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 18218* | Lemma 9 for funcsetcestrc 18219. (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 18219* | 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 18220* | 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 18221* | 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 18222* | 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 18223 | Extend class notation with the product of two categories. |
class ×c | ||
Syntax | c1stf 18224 | Extend class notation with the first projection functor. |
class 1stF | ||
Syntax | c2ndf 18225 | Extend class notation with the second projection functor. |
class 2ndF | ||
Syntax | cprf 18226 | Extend class notation with the functor pairing operation. |
class 〈,〉F | ||
Definition | df-xpc 18227* | 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 18228* | 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 18229* | 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 18230* | 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 18231 | The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.) |
⊢ ×c Fn (V × V) | ||
Theorem | xpcval 18232* | 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 18233 | 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 18234* | 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 18235 | 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 18236 | 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 18237* | 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 18238 | 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 18239 | 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 18240 | 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 18241 | 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 18242 | 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 18243* | 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 18244 | 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 18245 | 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 18246* | 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 18247 | 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 18248 | 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 18249* | 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 18250 | 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 18251 | 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 18252 | 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 18253 | 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 18254* | 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 18255 | 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 18256* | 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 18257 | 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 18258 | 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 18259 | 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 18260 | 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 18261 | 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 18262 | 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 | catcxpcclOLD 18263 | Obsolete version of catcxpccl 18262 as of 14-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 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 18264 | 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 18265 | Extend class notation with the evaluation functor. |
class evalF | ||
Syntax | ccurf 18266 | Extend class notation with the currying of a functor. |
class curryF | ||
Syntax | cuncf 18267 | Extend class notation with the uncurrying of a functor. |
class uncurryF | ||
Syntax | cdiag 18268 | Extend class notation to include the diagonal functor. |
class Δfunc | ||
Definition | df-evlf 18269* | 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 18270* | 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 18271* | 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 18272* | 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 18273* | 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 18274* | 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 18275 | 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 18276 | 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 18277 | Lemma for evlfcl 18278. (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 18278 | 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 18279* | 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 18280* | 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 18281* | 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 18282 | 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 18283 | 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 18284 | 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 18285* | 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 18286 | 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 18287 | 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 18288 | 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 18289 | 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 18290 | 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 18291 | 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 18292 | 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 18293 | 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 18294 | 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 18295 | 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 18296 | 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 18297 | 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 18298 | 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 18299 | 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 18300 | 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)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |