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Theorem List for Metamath Proof Explorer - 16701-16800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcurfuncf 16701 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 16702 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 16703 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 16704 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 16705 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 16706 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 16707 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 16708 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 16709 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 16710 As shown in diagval 16703, 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 16711 Extend class notation with the Hom functor.
class HomF

Syntaxcyon 16712 Extend class notation with the Yoneda embedding.
class Yon

Definitiondf-hof 16713* 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 16714 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 16715* 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 16716 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 16717 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 16718* 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 16719* 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 16720 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 16721 Lemma for hofcl 16722. (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 16722 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 16723 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 16724 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 16725 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 16726 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 16727 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 16728 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 16729 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 16730 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 16731 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 16732 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 16733 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 16734 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 16735 Lemma for yoneda 16746. (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 16736 Lemma for yoneda 16746. (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 16737* Lemma for yoneda 16746. (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 16738* Lemma for yoneda 16746. (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 16739* Lemma for yoneda 16746. (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 16740* Lemma for yoneda 16746. (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 16741 Lemma for yoneda 16746. (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 16742* Lemma for yoneda 16746. (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 16743* Lemma for yoneda 16746. (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 16744* 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 16745* Lemma for yonffth 16747. (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 16746* 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 16747 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 16748* 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 16749 Extend class notation with the class of all presets.
class Preset

Syntaxcdrs 16750 Extend class notation with the class of all directed sets.
class Dirset

Definitiondf-preset 16751* 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 16752* 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 16753* 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 16754 Lemma for prsref 16755 and prstr 16756. (Contributed by Mario Carneiro, 1-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Preset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))

Theoremprsref 16755 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 16756 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 16757* 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 16758* 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 16759 A directed set is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ Dirset → 𝐾 ∈ Preset )

Theoremdrsbn0 16760 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐵 = (Base‘𝐾)       (𝐾 ∈ Dirset → 𝐵 ≠ ∅)

Theoremdrsdirfi 16761* Any finite number of elements in a directed set have a common upper bound. Here is where the non-emptiness constraint in df-drs 16752 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 16762* 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 16763 Extend class notation with the class of posets.
class Poset

Syntaxcplt 16764 Extend class notation with less-than for posets.
class lt

Syntaxclub 16765 Extend class notation with poset least upper bound.
class lub

Syntaxcglb 16766 Extend class notation with poset greatest lower bound.
class glb

Syntaxcjn 16767 Extend class notation with poset join.
class join

Syntaxcmee 16768 Extend class notation with poset meet.
class meet

Definitiondf-poset 16769* 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 3218 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 16770* 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 16771* 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 16772 A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐾 ∈ Poset → 𝐾 ∈ Preset )

Theoremposi 16773 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 16774 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 16775 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 16776 A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))

Theorem0pos 16777 Technical lemma to simplify the statement of ipopos 16983. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 15739) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
∅ ∈ Poset

Theoremisposd 16778* Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.)
(𝜑𝐾 ∈ V)    &   (𝜑𝐵 = (Base‘𝐾))    &   (𝜑 = (le‘𝐾))    &   ((𝜑𝑥𝐵) → 𝑥 𝑥)    &   ((𝜑𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))       (𝜑𝐾 ∈ Poset)

Theoremisposi 16779* Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)
𝐾 ∈ V    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   (𝑥𝐵𝑥 𝑥)    &   ((𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))    &   ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))       𝐾 ∈ Poset

Theoremisposix 16780* Properties that determine a poset (explicit structure version). Note that the numeric indices of the structure components are not mentioned explicitly in either the theorem or its proof. (Contributed by NM, 9-Nov-2012.)
𝐵 ∈ V    &    ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ⟩}    &   (𝑥𝐵𝑥 𝑥)    &   ((𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))    &   ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))       𝐾 ∈ Poset

Definitiondf-plt 16781 Define less-than ordering for posets and related structures. Unlike df-base 15700 and df-ple 15788, this is a derived component extractor and not an extensible structure component extractor that defines the poset. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))

Theorempltfval 16782 Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
= (le‘𝐾)    &    < = (lt‘𝐾)       (𝐾𝐴< = ( ∖ I ))

Theorempltval 16783 Less-than relation. (df-pss 3556 analog.) (Contributed by NM, 12-Oct-2011.)
= (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))

Theorempltle 16784 Less-than implies less-than-or-equal. (pssss 3664 analog.) (Contributed by NM, 4-Dec-2011.)
= (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌𝑋 𝑌))

Theorempltne 16785 Less-than relation. (df-pss 3556 analog.) (Contributed by NM, 2-Dec-2011.)
< = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌𝑋𝑌))

Theorempltirr 16786 The less-than relation is not reflexive. (pssirr 3669 analog.) (Contributed by NM, 7-Feb-2012.)
< = (lt‘𝐾)       ((𝐾𝐴𝑋𝐵) → ¬ 𝑋 < 𝑋)

Theorempleval2i 16787 One direction of pleval2 16788. (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))

Theorempleval2 16788 Less-than-or-equal in terms of less-than. (sspss 3668 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))

Theorempltnle 16789 Less-than implies not inverse less-than-or-equal. (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 𝑋)

Theorempltval3 16790 Alternate expression for less-than relation. (dfpss3 3655 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))

Theorempltnlt 16791 The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 < 𝑋)

Theorempltn2lp 16792 The less-than relation has no 2-cycle loops. (pssn2lp 3670 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))

Theoremplttr 16793 The less-than relation is transitive. (psstr 3673 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))

Theorempltletr 16794 Transitive law for chained less-than and less-than-or-equal. (psssstr 3675 analog.) (Contributed by NM, 2-Dec-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))

Theoremplelttr 16795 Transitive law for chained less-than-or-equal and less-than. (sspsstr 3674 analog.) (Contributed by NM, 2-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))

Theorempospo 16796 Write a poset structure in terms of the proper-class poset predicate (strict less than version). (Contributed by Mario Carneiro, 8-Feb-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))

Definitiondf-lub 16797* Define the least upper bound (LUB) of a set of (poset) elements. The domain is restricted to exclude sets 𝑠 for which the LUB doesn't exist uniquely. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
lub = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑧𝑥(le‘𝑝)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑦(le‘𝑝)𝑧𝑥(le‘𝑝)𝑧))}))

Definitiondf-glb 16798* Define the greatest lower bound (GLB) of a set of (poset) elements. The domain is restricted to exclude sets 𝑠 for which the GLB doesn't exist uniquely. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)
glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑧(le‘𝑝)𝑦𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦𝑠 𝑧(le‘𝑝)𝑦𝑧(le‘𝑝)𝑥))}))

Definitiondf-join 16799* Define poset join. (Contributed by NM, 12-Sep-2011.) (Revised by Mario Carneiro, 3-Nov-2015.)
join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})

Definitiondf-meet 16800* Define poset join. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 8-Sep-2018.)
meet = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧})

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