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Theorem List for Metamath Proof Explorer - 16901-17000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-curf 16901* 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 16902* 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 16903* 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 16904* 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 16905* 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 16906 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 16907 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 16908 Lemma for evlfcl 16909. (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 16909 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 16910* 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 16911* 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 16912* 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 16913 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 16914 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 16915 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 16916* 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 16917 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 16918 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 16919 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 16920 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 16921 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 16922 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 16923 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 16924 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 16925 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 16926 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 16927 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 16928 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 16929 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 16930 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 16931 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 16932 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 16933 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 16934 As shown in diagval 16927, 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 16935 Extend class notation with the Hom functor.
class HomF

Syntaxcyon 16936 Extend class notation with the Yoneda embedding.
class Yon

Definitiondf-hof 16937* 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 16938 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 16939* 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 16940 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 16941 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 16942* 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 16943* 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 16944 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 16945 Lemma for hofcl 16946. (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 16946 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 16947 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 16948 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 16949 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 16950 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 16951 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 16952 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 16953 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 16954 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 16955 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 16956 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 16957 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 16958 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 16959 Lemma for yoneda 16970. (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 16960 Lemma for yoneda 16970. (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 16961* Lemma for yoneda 16970. (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)))

Theoremyonedalem4a 16962* Lemma for yoneda 16970. (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)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))    &   (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))       (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))

Theoremyonedalem4b 16963* Lemma for yoneda 16970. (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)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))    &   (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))    &   (𝜑𝑃𝐵)    &   (𝜑𝐺 ∈ (𝑃(Hom ‘𝐶)𝑋))       (𝜑 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑃)‘𝐺) = (((𝑋(2nd𝐹)𝑃)‘𝐺)‘𝐴))

Theoremyonedalem4c 16964* Lemma for yoneda 16970. (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)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))    &   (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))       (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Theoremyonedalem22 16965 Lemma for yoneda 16970. (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)    &   (𝜑𝐺 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑃𝐵)    &   (𝜑𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))    &   (𝜑𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))       (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd𝑌)𝑋)‘𝐾)(⟨((1st𝑌)‘𝑋), 𝐹⟩(2nd𝐻)⟨((1st𝑌)‘𝑃), 𝐺⟩)𝐴))

Theoremyonedalem3b 16966* Lemma for yoneda 16970. (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)    &   (𝜑𝐺 ∈ (𝑂 Func 𝑆))    &   (𝜑𝑃𝐵)    &   (𝜑𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))    &   (𝜑𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))       (𝜑 → ((𝐺𝑀𝑃)(⟨(𝐹(1st𝑍)𝑋), (𝐺(1st𝑍)𝑃)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐴(⟨𝐹, 𝑋⟩(2nd𝑍)⟨𝐺, 𝑃⟩)𝐾)) = ((𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑃⟩)𝐾)(⟨(𝐹(1st𝑍)𝑋), (𝐹(1st𝐸)𝑋)⟩(comp‘𝑇)(𝐺(1st𝐸)𝑃))(𝐹𝑀𝑋)))

Theoremyonedalem3 16967* Lemma for yoneda 16970. (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)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))       (𝜑𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))

Theoremyonedainv 16968* The Yoneda Lemma with explicit inverse. (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)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))    &   𝐼 = (Inv‘𝑅)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))       (𝜑𝑀(𝑍𝐼𝐸)𝑁)

Theoremyonffthlem 16969* Lemma for yonffth 16971. (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)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))    &   𝐼 = (Inv‘𝑅)    &   𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))       (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Theoremyoneda 16970* The Yoneda Lemma. There is a natural isomorphism between the functors 𝑍 and 𝐸, where 𝑍(𝐹, 𝑋) is the natural transformations from Yon(𝑋) = Hom ( − , 𝑋) to 𝐹, and 𝐸(𝐹, 𝑋) = 𝐹(𝑋) is the evaluation functor. Here we need two universes to state the claim: the smaller universe 𝑈 is used for forming the functor category 𝑄 = 𝐶 op → SetCat(𝑈), which itself does not (necessarily) live in 𝑈 but instead is an element of the larger universe 𝑉. (If 𝑈 is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set 𝑈 = 𝑉 in this case.) (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)    &   𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑎 ∈ (((1st𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎𝑥)‘( 1𝑥))))    &   𝐼 = (Iso‘𝑅)       (𝜑𝑀 ∈ (𝑍𝐼𝐸))

Theoremyonffth 16971 The Yoneda Lemma. The Yoneda embedding, the curried Hom functor, is full and faithful, and hence is a representation of the category 𝐶 as a full subcategory of the category 𝑄 of presheaves on 𝐶. (Contributed by Mario Carneiro, 29-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   (𝜑𝐶 ∈ Cat)    &   (𝜑𝑈𝑉)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)       (𝜑𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Theoremyoniso 16972* If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from 𝐶 into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)
𝑌 = (Yon‘𝐶)    &   𝑂 = (oppCat‘𝐶)    &   𝑆 = (SetCat‘𝑈)    &   𝐷 = (CatCat‘𝑉)    &   𝐵 = (Base‘𝐷)    &   𝐼 = (Iso‘𝐷)    &   𝑄 = (𝑂 FuncCat 𝑆)    &   𝐸 = (𝑄s ran (1st𝑌))    &   (𝜑𝑉𝑋)    &   (𝜑𝐶𝐵)    &   (𝜑𝑈𝑊)    &   (𝜑 → ran (Homf𝐶) ⊆ 𝑈)    &   (𝜑𝐸𝐵)    &   ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝐹‘(𝑥(Hom ‘𝐶)𝑦)) = 𝑦)       (𝜑𝑌 ∈ (𝐶𝐼𝐸))

PART 9  BASIC ORDER THEORY

9.1  Presets and directed sets using extensible structures

Syntaxcpreset 16973 Extend class notation with the class of all presets.
class Preset

Syntaxcdrs 16974 Extend class notation with the class of all directed sets.
class Dirset

Definitiondf-preset 16975* Define the class of preordered sets (presets). A preset is a set equipped with a transitive and reflexive relation.

Preorders are a natural generalization of order for sets where there is a well-defined ordering, but it in some sense "fails to capture the whole story", in that there may be pairs of elements which are indistinguishable under the order. Two elements which are not equal but are less-or-equal to each other behave the same under all order operations and may be thought of as "tied".

A preorder can naturally be strengthened by requiring that there are no ties, resulting in a partial order, or by stating that all comparable pairs of elements are tied, resulting in an equivalence relation. Every preorder naturally factors into these two types; the tied relation on a preorder is an equivalence relation and the quotient under that relation is a partial order. (Contributed by FL, 17-Nov-2014.) (Revised by Stefan O'Rear, 31-Jan-2015.)

Preset = {𝑓[(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}

Definitiondf-drs 16976* Define the class of directed sets. A directed set is a nonempty preordered set where every pair of elements have some upper bound. Note that it is not required that there exist a least upper bound.

There is no consensus in the literature over whether directed sets are allowed to be empty. It is slightly more convenient for us if they are not. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Dirset = {𝑓 ∈ Preset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}

Theoremisprs 16977* Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Preset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))

Theoremprslem 16978 Lemma for prsref 16979 and prstr 16980. (Contributed by Mario Carneiro, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Preset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))

Theoremprsref 16979 Less-or-equal is reflexive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Preset ∧ 𝑋𝐵) → 𝑋 𝑋)

Theoremprstr 16980 Less-or-equal is transitive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Preset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)

Theoremisdrs 16981* Property of being a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Dirset ↔ (𝐾 ∈ Preset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑧𝑦 𝑧)))

Theoremdrsdir 16982* Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Dirset ∧ 𝑋𝐵𝑌𝐵) → ∃𝑧𝐵 (𝑋 𝑧𝑌 𝑧))

Theoremdrsprs 16983 A directed set is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ Dirset → 𝐾 ∈ Preset )

Theoremdrsbn0 16984 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)       (𝐾 ∈ Dirset → 𝐵 ≠ ∅)

Theoremdrsdirfi 16985* Any finite number of elements in a directed set have a common upper bound. Here is where the nonemptiness constraint in df-drs 16976 first comes into play; without it we would need an additional constraint that 𝑋 not be empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Dirset ∧ 𝑋𝐵𝑋 ∈ Fin) → ∃𝑦𝐵𝑧𝑋 𝑧 𝑦)

Theoremisdrs2 16986* Directed sets may be defined in terms of finite subsets. Again, without nonemptiness we would need to restrict to nonempty subsets here. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Dirset ↔ (𝐾 ∈ Preset ∧ ∀𝑥 ∈ (𝒫 𝐵 ∩ Fin)∃𝑦𝐵𝑧𝑥 𝑧 𝑦))

9.2  Posets and lattices using extensible structures

9.2.1  Posets

Syntaxcpo 16987 Extend class notation with the class of posets.
class Poset

Syntaxcplt 16988 Extend class notation with less-than for posets.
class lt

Syntaxclub 16989 Extend class notation with poset least upper bound.
class lub

Syntaxcglb 16990 Extend class notation with poset greatest lower bound.
class glb

Syntaxcjn 16991 Extend class notation with poset join.
class join

Syntaxcmee 16992 Extend class notation with poset meet.
class meet

Definitiondf-poset 16993* Define the class of partially ordered sets (posets). A poset is a set equipped with a partial order, that is, a binary relation which is reflexive, antisymmetric, and transitive. Unlike a total order, in a partial order there may be pairs of elements where neither precedes the other. Definition of poset in [Crawley] p. 1. Note that Crawley-Dilworth require that a poset base set be nonempty, but we follow the convention of most authors who don't make this a requirement.

In our formalism of extensible structures, the base set of a poset 𝑓 is denoted by (Base‘𝑓) and its partial order by (le‘𝑓) (for "less than or equal to"). The quantifiers 𝑏𝑟 provide a notational shorthand to allow us to refer to the base and ordering relation as 𝑏 and 𝑟 in the definition rather than having to repeat (Base‘𝑓) and (le‘𝑓) throughout. These quantifiers can be eliminated with ceqsex2v 3276 and related theorems. (Contributed by NM, 18-Oct-2012.)

Poset = {𝑓 ∣ ∃𝑏𝑟(𝑏 = (Base‘𝑓) ∧ 𝑟 = (le‘𝑓) ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))}

Theoremispos 16994* The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))

Theoremispos2 16995* A poset is an antisymmetric preset.

EDITORIAL: could become the definition of poset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       (𝐾 ∈ Poset ↔ (𝐾 ∈ Preset ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))

Theoremposprs 16996 A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ Poset → 𝐾 ∈ Preset )

Theoremposi 16997 Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))

Theoremposref 16998 A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)

Theoremposasymb 16999 A poset ordering is asymmetric. (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))

Theorempostr 17000 A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))

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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 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