Home Metamath Proof ExplorerTheorem List (p. 173 of 437) < 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: Metamath Proof Explorer (1-28361) Hilbert Space Explorer (28362-29886) Users' Mathboxes (29887-43649)

Theorem List for Metamath Proof Explorer - 17201-17300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-prf 17201* Define the pairing operation for functors (which takes two functors 𝐹:𝐶𝐷 and 𝐺:𝐶𝐸 and produces (𝐹 ⟨,⟩F 𝐺):𝐶⟶(𝐷 ×c 𝐸)). (Contributed by Mario Carneiro, 11-Jan-2017.)
⟨,⟩F = (𝑓 ∈ V, 𝑔 ∈ V ↦ dom (1st𝑓) / 𝑏⟨(𝑥𝑏 ↦ ⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩), (𝑥𝑏, 𝑦𝑏 ↦ ( ∈ dom (𝑥(2nd𝑓)𝑦) ↦ ⟨((𝑥(2nd𝑓)𝑦)‘), ((𝑥(2nd𝑔)𝑦)‘)⟩))⟩)

Theoremfnxpc 17202 The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
×c Fn (V × V)

Theoremxpcval 17203* 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)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐵 = (𝑋 × 𝑌))    &   (𝜑𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣)))))    &   (𝜑𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩)))       (𝜑𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})

Theoremxpcbas 17204 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)

Theoremxpchomfval 17205* 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)       𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))

Theoremxpchom 17206 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)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))

Theoremrelxpchom 17207 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)

Theoremxpccofval 17208* 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)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))

Theoremxpcco 17209 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)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐾𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐾𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ (2nd𝑍))(2nd𝐹))⟩)

Theoremxpcco1st 17210 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)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐾𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐾𝑍))    &    · = (comp‘𝐶)       (𝜑 → (1st ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))

Theoremxpcco2nd 17211 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)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐾𝑌))    &   (𝜑𝐺 ∈ (𝑌𝐾𝑍))    &    · = (comp‘𝐷)       (𝜑 → (2nd ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩ · (2nd𝑍))(2nd𝐹)))

Theoremxpchom2 17212 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)    &   (𝜑𝑀𝑋)    &   (𝜑𝑁𝑌)    &   (𝜑𝑃𝑋)    &   (𝜑𝑄𝑌)    &   𝐾 = (Hom ‘𝑇)       (𝜑 → (⟨𝑀, 𝑁𝐾𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄)))

Theoremxpcco2 17213 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)    &   (𝜑𝑀𝑋)    &   (𝜑𝑁𝑌)    &   (𝜑𝑃𝑋)    &   (𝜑𝑄𝑌)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   𝑂 = (comp‘𝑇)    &   (𝜑𝑅𝑋)    &   (𝜑𝑆𝑌)    &   (𝜑𝐹 ∈ (𝑀𝐻𝑃))    &   (𝜑𝐺 ∈ (𝑁𝐽𝑄))    &   (𝜑𝐾 ∈ (𝑃𝐻𝑅))    &   (𝜑𝐿 ∈ (𝑄𝐽𝑆))       (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃· 𝑅)𝐹), (𝐿(⟨𝑁, 𝑄 𝑆)𝐺)⟩)

Theoremxpccatid 17214* The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)    &   𝐼 = (Id‘𝐶)    &   𝐽 = (Id‘𝐷)       (𝜑 → (𝑇 ∈ Cat ∧ (Id‘𝑇) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝐼𝑥), (𝐽𝑦)⟩)))

Theoremxpcid 17215 The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑋 = (Base‘𝐶)    &   𝑌 = (Base‘𝐷)    &   𝐼 = (Id‘𝐶)    &   𝐽 = (Id‘𝐷)    &    1 = (Id‘𝑇)    &   (𝜑𝑅𝑋)    &   (𝜑𝑆𝑌)       (𝜑 → ( 1 ‘⟨𝑅, 𝑆⟩) = ⟨(𝐼𝑅), (𝐽𝑆)⟩)

Theoremxpccat 17216 The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑𝑇 ∈ Cat)

Theorem1stfval 17217* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝑇)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑃 = (𝐶 1stF 𝐷)       (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)

Theorem1stf1 17218 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑃 = (𝐶 1stF 𝐷)    &   (𝜑𝑅𝐵)       (𝜑 → ((1st𝑃)‘𝑅) = (1st𝑅))

Theorem1stf2 17219 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑃 = (𝐶 1stF 𝐷)    &   (𝜑𝑅𝐵)    &   (𝜑𝑆𝐵)       (𝜑 → (𝑅(2nd𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))

Theorem2ndfval 17220* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝑇)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐶 2ndF 𝐷)       (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)

Theorem2ndf1 17221 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐶 2ndF 𝐷)    &   (𝜑𝑅𝐵)       (𝜑 → ((1st𝑄)‘𝑅) = (2nd𝑅))

Theorem2ndf2 17222 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐶 2ndF 𝐷)    &   (𝜑𝑅𝐵)    &   (𝜑𝑆𝐵)       (𝜑 → (𝑅(2nd𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))

Theorem1stfcl 17223 The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑃 = (𝐶 1stF 𝐷)       (𝜑𝑃 ∈ (𝑇 Func 𝐶))

Theorem2ndfcl 17224 The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
𝑇 = (𝐶 ×c 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐶 2ndF 𝐷)       (𝜑𝑄 ∈ (𝑇 Func 𝐷))

Theoremprfval 17225* Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))       (𝜑𝑃 = ⟨(𝑥𝐵 ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥𝐵, 𝑦𝐵 ↦ ( ∈ (𝑥𝐻𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)

Theoremprf1 17226 Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑𝑋𝐵)       (𝜑 → ((1st𝑃)‘𝑋) = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑋)⟩)

Theoremprf2fval 17227* Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋(2nd𝑃)𝑌) = ( ∈ (𝑋𝐻𝑌) ↦ ⟨((𝑋(2nd𝐹)𝑌)‘), ((𝑋(2nd𝐺)𝑌)‘)⟩))

Theoremprf2 17228 Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐾 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝑋(2nd𝑃)𝑌)‘𝐾) = ⟨((𝑋(2nd𝐹)𝑌)‘𝐾), ((𝑋(2nd𝐺)𝑌)‘𝐾)⟩)

Theoremprfcl 17229 The pairing of functors 𝐹:𝐶𝐷 and 𝐺:𝐶𝐷 is a functor 𝐹, 𝐺⟩:𝐶⟶(𝐷 × 𝐸). (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   𝑇 = (𝐷 ×c 𝐸)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))       (𝜑𝑃 ∈ (𝐶 Func 𝑇))

Theoremprf1st 17230 Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))       (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)

Theoremprf2nd 17231 Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑃 = (𝐹 ⟨,⟩F 𝐺)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐸))       (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝑃) = 𝐺)

Theorem1st2ndprf 17232 Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝑇 = (𝐷 ×c 𝐸)    &   (𝜑𝐹 ∈ (𝐶 Func 𝑇))    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)       (𝜑𝐹 = (((𝐷 1stF 𝐸) ∘func 𝐹) ⟨,⟩F ((𝐷 2ndF 𝐸) ∘func 𝐹)))

Theoremcatcxpccl 17233 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)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑𝑇𝐵)

Theoremxpcpropd 17234 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 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → (𝐴 ×c 𝐶) = (𝐵 ×c 𝐷))

8.4.2  Functor evaluation

Syntaxcevlf 17235 Extend class notation with the evaluation functor.
class evalF

Syntaxccurf 17236 Extend class notation with the currying of a functor.
class curryF

Syntaxcuncf 17237 Extend class notation with the uncurrying of a functor.
class uncurryF

Syntaxcdiag 17238 Extend class notation to include the diagonal functor.
class Δfunc

Definitiondf-evlf 17239* Define the evaluation functor, which is the extension of the evaluation map 𝑓, 𝑥 ↦ (𝑓𝑥) of functors, to a functor (𝐶𝐷) × 𝐶𝐷. (Contributed by Mario Carneiro, 11-Jan-2017.)
evalF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⟨(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩(comp‘𝑑)((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)

Definitiondf-curf 17240* Define the curry functor, which maps a functor 𝐹:𝐶 × 𝐷𝐸 to curryF (𝐹):𝐶⟶(𝐷𝐸). (Contributed by Mario Carneiro, 11-Jan-2017.)
curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦ (1st𝑒) / 𝑐(2nd𝑒) / 𝑑⟨(𝑥 ∈ (Base‘𝑐) ↦ ⟨(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(⟨𝑥, 𝑦⟩(2nd𝑓)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝑓)⟨𝑦, 𝑧⟩)((Id‘𝑑)‘𝑧)))))⟩)

Definitiondf-uncf 17241* 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 ((𝑓func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))

Definitiondf-diag 17242* Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥𝐶 ↦ (𝑦𝐷𝑥). 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:𝑥𝑦 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)))

Theoremevlfval 17243* Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = (𝐶 evalF 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐷)    &   𝑁 = (𝐶 Nat 𝐷)       (𝜑𝐸 = ⟨(𝑓 ∈ (𝐶 Func 𝐷), 𝑥𝐵 ↦ ((1st𝑓)‘𝑥)), (𝑥 ∈ ((𝐶 Func 𝐷) × 𝐵), 𝑦 ∈ ((𝐶 Func 𝐷) × 𝐵) ↦ (1st𝑥) / 𝑚(1st𝑦) / 𝑛(𝑎 ∈ (𝑚𝑁𝑛), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ((𝑎‘(2nd𝑦))(⟨((1st𝑚)‘(2nd𝑥)), ((1st𝑚)‘(2nd𝑦))⟩ · ((1st𝑛)‘(2nd𝑦)))(((2nd𝑥)(2nd𝑚)(2nd𝑦))‘𝑔))))⟩)

Theoremevlf2 17244* Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = (𝐶 evalF 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐿 = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)       (𝜑𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝑔))))

Theoremevlf2val 17245 Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = (𝐶 evalF 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝐺 ∈ (𝐶 Func 𝐷))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐿 = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)    &   (𝜑𝐴 ∈ (𝐹𝑁𝐺))    &   (𝜑𝐾 ∈ (𝑋𝐻𝑌))       (𝜑 → (𝐴𝐿𝐾) = ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩ · ((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)))

Theoremevlf1 17246 Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = (𝐶 evalF 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑𝑋𝐵)       (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))

Theoremevlfcllem 17247 Lemma for evlfcl 17248. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = (𝐶 evalF 𝐷)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ 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⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = (((⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)(⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨𝐻, 𝑍⟩))((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)))

Theoremevlfcl 17248 The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors 𝐶𝐷, and the second parameter in 𝐷. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐸 = (𝐶 evalF 𝐷)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑𝐸 ∈ ((𝑄 ×c 𝐶) Func 𝐷))

Theoremcurfval 17249* Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   𝐽 = (Hom ‘𝐷)    &    1 = (Id‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Id‘𝐷)       (𝜑𝐺 = ⟨(𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩), (𝑥𝐴, 𝑦𝐴 ↦ (𝑔 ∈ (𝑥𝐻𝑦) ↦ (𝑧𝐵 ↦ (𝑔(⟨𝑥, 𝑧⟩(2nd𝐹)⟨𝑦, 𝑧⟩)(𝐼𝑧)))))⟩)

Theoremcurf1fval 17250* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   𝐽 = (Hom ‘𝐷)    &    1 = (Id‘𝐶)       (𝜑 → (1st𝐺) = (𝑥𝐴 ↦ ⟨(𝑦𝐵 ↦ (𝑥(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑥)(⟨𝑥, 𝑦⟩(2nd𝐹)⟨𝑥, 𝑧⟩)𝑔)))⟩))

Theoremcurf1 17251* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   (𝜑𝑋𝐴)    &   𝐾 = ((1st𝐺)‘𝑋)    &   𝐽 = (Hom ‘𝐷)    &    1 = (Id‘𝐶)       (𝜑𝐾 = ⟨(𝑦𝐵 ↦ (𝑋(1st𝐹)𝑦)), (𝑦𝐵, 𝑧𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (( 1𝑋)(⟨𝑋, 𝑦⟩(2nd𝐹)⟨𝑋, 𝑧⟩)𝑔)))⟩)

Theoremcurf11 17252 Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   (𝜑𝑋𝐴)    &   𝐾 = ((1st𝐺)‘𝑋)    &   (𝜑𝑌𝐵)       (𝜑 → ((1st𝐾)‘𝑌) = (𝑋(1st𝐹)𝑌))

Theoremcurf12 17253 The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   (𝜑𝑋𝐴)    &   𝐾 = ((1st𝐺)‘𝑋)    &   (𝜑𝑌𝐵)    &   𝐽 = (Hom ‘𝐷)    &    1 = (Id‘𝐶)    &   (𝜑𝑍𝐵)    &   (𝜑𝐻 ∈ (𝑌𝐽𝑍))       (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐻) = (( 1𝑋)(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑋, 𝑍⟩)𝐻))

Theoremcurf1cl 17254 The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   (𝜑𝑋𝐴)    &   𝐾 = ((1st𝐺)‘𝑋)       (𝜑𝐾 ∈ (𝐷 Func 𝐸))

Theoremcurf2 17255* Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Id‘𝐷)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝐾 ∈ (𝑋𝐻𝑌))    &   𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)       (𝜑𝐿 = (𝑧𝐵 ↦ (𝐾(⟨𝑋, 𝑧⟩(2nd𝐹)⟨𝑌, 𝑧⟩)(𝐼𝑧))))

Theoremcurf2val 17256 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Id‘𝐷)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝐾 ∈ (𝑋𝐻𝑌))    &   𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)    &   (𝜑𝑍𝐵)       (𝜑 → (𝐿𝑍) = (𝐾(⟨𝑋, 𝑍⟩(2nd𝐹)⟨𝑌, 𝑍⟩)(𝐼𝑍)))

Theoremcurf2cl 17257 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))    &   𝐵 = (Base‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Id‘𝐷)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝐾 ∈ (𝑋𝐻𝑌))    &   𝐿 = ((𝑋(2nd𝐺)𝑌)‘𝐾)    &   𝑁 = (𝐷 Nat 𝐸)       (𝜑𝐿 ∈ (((1st𝐺)‘𝑋)𝑁((1st𝐺)‘𝑌)))

Theoremcurfcl 17258 The curry functor of a functor 𝐹:𝐶 × 𝐷𝐸 is a functor curryF (𝐹):𝐶⟶(𝐷𝐸). (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   𝑄 = (𝐷 FuncCat 𝐸)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))       (𝜑𝐺 ∈ (𝐶 Func 𝑄))

Theoremcurfpropd 17259 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 → (Homf𝐴) = (Homf𝐵))    &   (𝜑 → (compf𝐴) = (compf𝐵))    &   (𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐴 ∈ Cat)    &   (𝜑𝐵 ∈ Cat)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐴 ×c 𝐶) Func 𝐸))       (𝜑 → (⟨𝐴, 𝐶⟩ curryF 𝐹) = (⟨𝐵, 𝐷⟩ curryF 𝐹))

Theoremuncfval 17260 Value of the uncurry functor, which is the reverse of the curry functor, taking 𝐺:𝐶⟶(𝐷𝐸) to uncurryF (𝐺):𝐶 × 𝐷𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)    &   (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))       (𝜑𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))

Theoremuncfcl 17261 The uncurry operation takes a functor 𝐹:𝐶⟶(𝐷𝐸) to a functor uncurryF (𝐹):𝐶 × 𝐷𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)    &   (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))       (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))

Theoremuncf1 17262 Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)    &   (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))    &   𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋(1st𝐹)𝑌) = ((1st ‘((1st𝐺)‘𝑋))‘𝑌))

Theoremuncf2 17263 Value of the uncurry functor on a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)    &   (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))    &   𝐴 = (Base‘𝐶)    &   𝐵 = (Base‘𝐷)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐵)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   (𝜑𝑍𝐴)    &   (𝜑𝑊𝐵)    &   (𝜑𝑅 ∈ (𝑋𝐻𝑍))    &   (𝜑𝑆 ∈ (𝑌𝐽𝑊))       (𝜑 → (𝑅(⟨𝑋, 𝑌⟩(2nd𝐹)⟨𝑍, 𝑊⟩)𝑆) = ((((𝑋(2nd𝐺)𝑍)‘𝑅)‘𝑊)(⟨((1st ‘((1st𝐺)‘𝑋))‘𝑌), ((1st ‘((1st𝐺)‘𝑋))‘𝑊)⟩(comp‘𝐸)((1st ‘((1st𝐺)‘𝑍))‘𝑊))((𝑌(2nd ‘((1st𝐺)‘𝑋))𝑊)‘𝑆)))

Theoremcurfuncf 17264 Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐸 ∈ Cat)    &   (𝜑𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))       (𝜑 → (⟨𝐶, 𝐷⟩ curryF 𝐹) = 𝐺)

Theoremuncfcurf 17265 Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.)
𝐺 = (⟨𝐶, 𝐷⟩ curryF 𝐹)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸))       (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = 𝐹)

Theoremdiagval 17266 Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥𝐶 ↦ (𝑦𝐷𝑥). 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑𝐿 = (⟨𝐶, 𝐷⟩ curryF (𝐶 1stF 𝐷)))

Theoremdiagcl 17267 The diagonal functor is a functor from the base category to the functor category. Another way of saying this is that the constant functor (𝑦𝐷𝑋) 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝑄 = (𝐷 FuncCat 𝐶)       (𝜑𝐿 ∈ (𝐶 Func 𝑄))

Theoremdiag1cl 17268 The constant functor of 𝑋 is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
𝐿 = (𝐶Δfunc𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝑋𝐴)    &   𝐾 = ((1st𝐿)‘𝑋)       (𝜑𝐾 ∈ (𝐷 Func 𝐶))

Theoremdiag11 17269 Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
𝐿 = (𝐶Δfunc𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝑋𝐴)    &   𝐾 = ((1st𝐿)‘𝑋)    &   𝐵 = (Base‘𝐷)    &   (𝜑𝑌𝐵)       (𝜑 → ((1st𝐾)‘𝑌) = 𝑋)

Theoremdiag12 17270 Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
𝐿 = (𝐶Δfunc𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   𝐴 = (Base‘𝐶)    &   (𝜑𝑋𝐴)    &   𝐾 = ((1st𝐿)‘𝑋)    &   𝐵 = (Base‘𝐷)    &   (𝜑𝑌𝐵)    &   𝐽 = (Hom ‘𝐷)    &    1 = (Id‘𝐶)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑌𝐽𝑍))       (𝜑 → ((𝑌(2nd𝐾)𝑍)‘𝐹) = ( 1𝑋))

Theoremdiag2 17271 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝑋(2nd𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))

Theoremdiag2cl 17272 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)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑌))    &   𝑁 = (𝐷 Nat 𝐶)       (𝜑 → (𝐵 × {𝐹}) ∈ (((1st𝐿)‘𝑋)𝑁((1st𝐿)‘𝑌)))

Theoremcurf2ndf 17273 As shown in diagval 17266, the currying of the first projection is the diagonal functor. On the other hand, the currying of the second projection is 𝑥𝐶 ↦ (𝑦𝐷𝑦), which is a constant functor of the identity functor at 𝐷. (Contributed by Mario Carneiro, 15-Jan-2017.)
𝑄 = (𝐷 FuncCat 𝐷)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (⟨𝐶, 𝐷⟩ curryF (𝐶 2ndF 𝐷)) = ((1st ‘(𝑄Δfunc𝐶))‘(idfunc𝐷)))

8.4.3  Hom functor

Syntaxchof 17274 Extend class notation with the Hom functor.
class HomF

Syntaxcyon 17275 Extend class notation with the Yoneda embedding.
class Yon

Definitiondf-hof 17276* 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 ↦ ⟨(Homf𝑐), (Base‘𝑐) / 𝑏(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝑐)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓))))⟩)

Definitiondf-yon 17277 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))))

Theoremhofval 17278* 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 = (HomF𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)       (𝜑𝑀 = ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩)

Theoremhof1fval 17279 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 = (HomF𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑 → (1st𝑀) = (Homf𝐶))

Theoremhof1 17280 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 = (HomF𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋(1st𝑀)𝑌) = (𝑋𝐻𝑌))

Theoremhof2fval 17281* 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:𝑍𝑋 and 𝑔:𝑌𝑊), yields a function (a morphism of SetCat) mapping :𝑋𝑌 to 𝑔𝑓:𝑍𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)
𝑀 = (HomF𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝐵)    &    · = (comp‘𝐶)       (𝜑 → (⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ ( ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝑓))))

Theoremhof2val 17282* 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:𝑍𝑋 and 𝑔:𝑌𝑊), yields a function (a morphism of SetCat) mapping :𝑋𝑌 to 𝑔𝑓:𝑍𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)
𝑀 = (HomF𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝐹 ∈ (𝑍𝐻𝑋))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑊))       (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐺) = ( ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌· 𝑊))(⟨𝑍, 𝑋· 𝑊)𝐹)))

Theoremhof2 17283 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:𝑍𝑋 and 𝑔:𝑌𝑊), yields a function (a morphism of SetCat) mapping :𝑋𝑌 to 𝑔𝑓:𝑍𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)
𝑀 = (HomF𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝐹 ∈ (𝑍𝐻𝑋))    &   (𝜑𝐺 ∈ (𝑌𝐻𝑊))    &   (𝜑𝐾 ∈ (𝑋𝐻𝑌))       (𝜑 → ((𝐹(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐺)‘𝐾) = ((𝐺(⟨𝑋, 𝑌· 𝑊)𝐾)(⟨𝑍, 𝑋· 𝑊)𝐹))

Theoremhofcllem 17284 Lemma for hofcl 17285. (Contributed by Mario Carneiro, 15-Jan-2017.)
𝑀 = (HomF𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝐷 = (SetCat‘𝑈)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝐵)    &   (𝜑𝑆𝐵)    &   (𝜑𝑇𝐵)    &   (𝜑𝐾 ∈ (𝑍𝐻𝑋))    &   (𝜑𝐿 ∈ (𝑌𝐻𝑊))    &   (𝜑𝑃 ∈ (𝑆𝐻𝑍))    &   (𝜑𝑄 ∈ (𝑊𝐻𝑇))       (𝜑 → ((𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑆, 𝑇⟩)(𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)) = ((𝑃(⟨𝑍, 𝑊⟩(2nd𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐿)))

Theoremhofcl 17285 Closure of the Hom functor. Note that the codomain is the category SetCat‘𝑈 for any universe 𝑈 which contains each Hom-set. This corresponds to the assertion that 𝐶 be locally small (with respect to 𝑈). (Contributed by Mario Carneiro, 15-Jan-2017.)
𝑀 = (HomF𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝐷 = (SetCat‘𝑈)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑𝑀 ∈ ((𝑂 ×c 𝐶) Func 𝐷))

Theoremoppchofcl 17286 Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑀 = (HomF𝑂)    &   𝐷 = (SetCat‘𝑈)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑𝑀 ∈ ((𝐶 ×c 𝑂) Func 𝐷))

Theoremyonval 17287 Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝑂 = (oppCat‘𝐶)    &   𝑀 = (HomF𝑂)       (𝜑𝑌 = (⟨𝐶, 𝑂⟩ curryF 𝑀))

Theoremyoncl 17288 The Yoneda embedding is a functor from the category to the category 𝑄 of presheaves on 𝐶. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑𝑌 ∈ (𝐶 Func 𝑄))

Theoremyon1cl 17289 The Yoneda embedding at an object of 𝐶 is a presheaf on 𝐶, also known as the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))

Theoremyon11 17290 Value of the Yoneda embedding at an object. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑍𝐵)       (𝜑 → ((1st ‘((1st𝑌)‘𝑋))‘𝑍) = (𝑍𝐻𝑋))

Theoremyon12 17291 Value of the Yoneda embedding at a morphism. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑍𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝑊𝐵)    &   (𝜑𝐹 ∈ (𝑊𝐻𝑍))    &   (𝜑𝐺 ∈ (𝑍𝐻𝑋))       (𝜑 → (((𝑍(2nd ‘((1st𝑌)‘𝑋))𝑊)‘𝐹)‘𝐺) = (𝐺(⟨𝑊, 𝑍· 𝑋)𝐹))

Theoremyon2 17292 Value of the Yoneda embedding at a morphism. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑋𝐵)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑍𝐵)    &    · = (comp‘𝐶)    &   (𝜑𝑊𝐵)    &   (𝜑𝐹 ∈ (𝑋𝐻𝑍))    &   (𝜑𝐺 ∈ (𝑊𝐻𝑋))       (𝜑 → ((((𝑋(2nd𝑌)𝑍)‘𝐹)‘𝑊)‘𝐺) = (𝐹(⟨𝑊, 𝑋· 𝑍)𝐺))

Theoremhofpropd 17293 If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (HomF𝐶) = (HomF𝐷))

Theoremyonpropd 17294 If two categories have the same set of objects, morphisms, and compositions, then they have the same Yoneda functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑 → (Homf𝐶) = (Homf𝐷))    &   (𝜑 → (compf𝐶) = (compf𝐷))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝐷 ∈ Cat)       (𝜑 → (Yon‘𝐶) = (Yon‘𝐷))

Theoremoppcyon 17295 Value of the opposite Yoneda embedding. (Contributed by Mario Carneiro, 26-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑌 = (Yon‘𝑂)    &   𝑀 = (HomF𝐶)    &   (𝜑𝐶 ∈ Cat)       (𝜑𝑌 = (⟨𝑂, 𝐶⟩ curryF 𝑀))

Theoremoyoncl 17296 The opposite Yoneda embedding is a functor from oppCat‘𝐶 to the functor category 𝐶 → SetCat. (Contributed by Mario Carneiro, 26-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑌 = (Yon‘𝑂)    &   (𝜑𝐶 ∈ Cat)    &   𝑆 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   𝑄 = (𝐶 FuncCat 𝑆)       (𝜑𝑌 ∈ (𝑂 Func 𝑄))

Theoremoyon1cl 17297 The opposite Yoneda embedding at an object of 𝐶 is a functor from 𝐶 to Set, also known as the covariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
𝑂 = (oppCat‘𝐶)    &   𝑌 = (Yon‘𝑂)    &   (𝜑𝐶 ∈ Cat)    &   𝑆 = (SetCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝐶 Func 𝑆))

Theoremyonedalem1 17298 Lemma for yoneda 17309. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)       (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))

Theoremyonedalem21 17299 Lemma for yoneda 17309. (Contributed by Mario Carneiro, 28-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)       (𝜑 → (𝐹(1st𝑍)𝑋) = (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Theoremyonedalem3a 17300* Lemma for yoneda 17309. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑇 = (SetCat‘𝑉)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐻 = (HomF𝑄)    &   𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)    &   𝐸 = (𝑂 evalF 𝑆)    &   𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑉𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)    &   (𝜑𝐹 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑋𝐵)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))       (𝜑 → ((𝐹𝑀𝑋) = (𝑎 ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↦ ((𝑎𝑋)‘( 1𝑋))) ∧ (𝐹𝑀𝑋):(𝐹(1st𝑍)𝑋)⟶(𝐹(1st𝐸)𝑋)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43649
 Copyright terms: Public domain < Previous  Next >