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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | uobeqw 49201 | If a full functor (in fact, a full embedding) is a section of a fully faithful functor (surjective on objects), then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺) & ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌) & ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸)) & ⊢ (𝜑 → (𝐿 ∘func 𝐾) = 𝐼) & ⊢ (𝜑 → 𝐿 ∈ ((𝐸 Full 𝐷) ∩ (𝐸 Faith 𝐷))) ⇒ ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌)) | ||
| Theorem | uobeq 49202 | If a full functor (in fact, a full embedding) is a section of a functor (surjective on objects), then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺) & ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌) & ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸)) & ⊢ (𝜑 → (𝐿 ∘func 𝐾) = 𝐼) & ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷)) ⇒ ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌)) | ||
| Theorem | uptr2 49203 | Universal property and fully faithful functor surjective on objects. (Contributed by Zhi Wang, 25-Nov-2025.) |
| ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑌 = (𝑅‘𝑋)) & ⊢ (𝜑 → 𝑅:𝐴–onto→𝐵) & ⊢ (𝜑 → 𝑅((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝑆) & ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝑅, 𝑆⟩) = ⟨𝐹, 𝐺⟩) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿) ⇒ ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀)) | ||
| Theorem | uptr2a 49204 | Universal property and fully faithful functor surjective on objects. (Contributed by Zhi Wang, 25-Nov-2025.) |
| ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑌 = ((1st ‘𝐾)‘𝑋)) & ⊢ (𝜑 → (𝐺 ∘func 𝐾) = 𝐹) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐾 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))) & ⊢ (𝜑 → (1st ‘𝐾):𝐴–onto→𝐵) ⇒ ⊢ (𝜑 → (𝑋(𝐹(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(𝐺(𝐷 UP 𝐸)𝑍)𝑀)) | ||
| Theorem | isnatd 49205* | Property of being a natural transformation; deduction form. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ · = (comp‘𝐷) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) & ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿) & ⊢ (𝜑 → 𝐴 Fn 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) & ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ℎ ∈ (𝑥𝐻𝑦)) → ((𝐴‘𝑦)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ · (𝐾‘𝑦))((𝑥𝐺𝑦)‘ℎ)) = (((𝑥𝐿𝑦)‘ℎ)(⟨(𝐹‘𝑥), (𝐾‘𝑥)⟩ · (𝐾‘𝑦))(𝐴‘𝑥))) ⇒ ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩)) | ||
| Theorem | natrcl2 49206 | Reverse closure for a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩)) ⇒ ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) | ||
| Theorem | natrcl3 49207 | Reverse closure for a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩)) ⇒ ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿) | ||
| Theorem | catbas 49208 | The base of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} & ⊢ 𝐵 ∈ V ⇒ ⊢ 𝐵 = (Base‘𝐶) | ||
| Theorem | cathomfval 49209 | The hom-sets of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} & ⊢ 𝐻 ∈ V ⇒ ⊢ 𝐻 = (Hom ‘𝐶) | ||
| Theorem | catcofval 49210 | Composition of the category structure. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} & ⊢ · ∈ V ⇒ ⊢ · = (comp‘𝐶) | ||
| Theorem | natoppf 49211 | A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ 𝑀 = (𝑂 Nat 𝑃) & ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩𝑁⟨𝐾, 𝐿⟩)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (⟨𝐾, tpos 𝐿⟩𝑀⟨𝐹, tpos 𝐺⟩)) | ||
| Theorem | natoppf2 49212 | A natural transformation is natural between opposite functors. (Contributed by Zhi Wang, 18-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ 𝑀 = (𝑂 Nat 𝑃) & ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹)) & ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐿𝑀𝐾)) | ||
| Theorem | natoppfb 49213 | A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ 𝑁 = (𝐶 Nat 𝐷) & ⊢ 𝑀 = (𝑂 Nat 𝑃) & ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹)) & ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾)) | ||
| Theorem | initoo2 49214 | An initial object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵) | ||
| Theorem | termoo2 49215 | A terminal object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ 𝐵) | ||
| Theorem | zeroo2 49216 | A zero object is an object in the base set. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑂 ∈ (ZeroO‘𝐶) → 𝑂 ∈ 𝐵) | ||
| Theorem | oppcinito 49217 | Initial objects are terminal in the opposite category. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ (TermO‘(oppCat‘𝐶))) | ||
| Theorem | oppctermo 49218 | Terminal objects are initial in the opposite category. Comments before Definition 7.4 in [Adamek] p. 102. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝐼 ∈ (TermO‘𝐶) ↔ 𝐼 ∈ (InitO‘(oppCat‘𝐶))) | ||
| Theorem | oppczeroo 49219 | Zero objects are zero in the opposite category. Remark 7.8 of [Adamek] p. 103. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐼 ∈ (ZeroO‘𝐶) ↔ 𝐼 ∈ (ZeroO‘(oppCat‘𝐶))) | ||
| Theorem | termoeu2 49220 | Terminal objects are essentially unique; if 𝐴 is a terminal object, then so is every object that is isomorphic to 𝐴. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶)) & ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶)) | ||
| Theorem | initopropdlemlem 49221 | Lemma for initopropdlem 49222, termopropdlem 49223, and zeroopropdlem 49224. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ 𝐹 Fn 𝑋 & ⊢ (𝜑 → ¬ 𝐴 ∈ 𝑌) & ⊢ 𝑋 ⊆ 𝑌 & ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵)) | ||
| Theorem | initopropdlem 49222 | Lemma for initopropd 49225. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → ¬ 𝐶 ∈ V) ⇒ ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷)) | ||
| Theorem | termopropdlem 49223 | Lemma for termopropd 49226. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → ¬ 𝐶 ∈ V) ⇒ ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷)) | ||
| Theorem | zeroopropdlem 49224 | Lemma for zeroopropd 49227. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → ¬ 𝐶 ∈ V) ⇒ ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷)) | ||
| Theorem | initopropd 49225 | Two structures with the same base, hom-sets and composition operation have the same initial objects. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (InitO‘𝐶) = (InitO‘𝐷)) | ||
| Theorem | termopropd 49226 | Two structures with the same base, hom-sets and composition operation have the same terminal objects. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷)) | ||
| Theorem | zeroopropd 49227 | Two structures with the same base, hom-sets and composition operation have the same zero objects. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (ZeroO‘𝐶) = (ZeroO‘𝐷)) | ||
| Theorem | reldmxpc 49228 | The binary product of categories is a proper operator, so it can be used with ovprc1 7408, elbasov 17162, strov2rcl 17163, and so on. See reldmxpcALT 49229 for an alternate proof with less "essential steps" but more "bytes". (Proposed by SN, 15-Oct-2025.) (Contributed by Zhi Wang, 15-Oct-2025.) |
| ⊢ Rel dom ×c | ||
| Theorem | reldmxpcALT 49229 | Alternate proof of reldmxpc 49228. (Contributed by Zhi Wang, 15-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Rel dom ×c | ||
| Theorem | elxpcbasex1 49230 | A non-empty base set of the product category indicates the existence of the first factor of the product category. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof shortened by SN, 15-Oct-2025.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ V) | ||
| Theorem | elxpcbasex1ALT 49231 | Alternate proof of elxpcbasex1 49230. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ V) | ||
| Theorem | elxpcbasex2 49232 | A non-empty base set of the product category indicates the existence of the second factor of the product category. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof shortened by SN, 15-Oct-2025.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐷 ∈ V) | ||
| Theorem | elxpcbasex2ALT 49233 | Alternate proof of elxpcbasex2 49232. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑇 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐷 ∈ V) | ||
| Theorem | xpcfucbas 49234 | The base set of the product of two categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) ⇒ ⊢ ((𝐵 Func 𝐶) × (𝐷 Func 𝐸)) = (Base‘𝑇) | ||
| Theorem | xpcfuchomfval 49235* | Set of morphisms of the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝐴 = (Base‘𝑇) & ⊢ 𝐾 = (Hom ‘𝑇) ⇒ ⊢ 𝐾 = (𝑢 ∈ 𝐴, 𝑣 ∈ 𝐴 ↦ (((1st ‘𝑢)(𝐵 Nat 𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(𝐷 Nat 𝐸)(2nd ‘𝑣)))) | ||
| Theorem | xpcfuchom 49236 | Set of morphisms of the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝐴 = (Base‘𝑇) & ⊢ 𝐾 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)(𝐵 Nat 𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(𝐷 Nat 𝐸)(2nd ‘𝑌)))) | ||
| Theorem | xpcfuchom2 49237 | Value of the set of morphisms in the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ (𝜑 → 𝑀 ∈ (𝐵 Func 𝐶)) & ⊢ (𝜑 → 𝑁 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝑃 ∈ (𝐵 Func 𝐶)) & ⊢ (𝜑 → 𝑄 ∈ (𝐷 Func 𝐸)) & ⊢ 𝐾 = (Hom ‘𝑇) ⇒ ⊢ (𝜑 → (⟨𝑀, 𝑁⟩𝐾⟨𝑃, 𝑄⟩) = ((𝑀(𝐵 Nat 𝐶)𝑃) × (𝑁(𝐷 Nat 𝐸)𝑄))) | ||
| Theorem | xpcfucco2 49238 | Value of composition in the binary product of categories of functors. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃)) & ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅)) & ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆)) ⇒ ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺)⟩) | ||
| Theorem | xpcfuccocl 49239 | The composition of two natural transformations is a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃)) & ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅)) & ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆)) ⇒ ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) ∈ ((𝑀(𝐵 Nat 𝐶)𝑅) × (𝑁(𝐷 Nat 𝐸)𝑆))) | ||
| Theorem | xpcfucco3 49240* | Value of composition in the binary product of categories of functors; expressed explicitly. (Contributed by Zhi Wang, 1-Oct-2025.) |
| ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×c (𝐷 FuncCat 𝐸)) & ⊢ 𝑂 = (comp‘𝑇) & ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃)) & ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄)) & ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅)) & ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆)) & ⊢ 𝑋 = (Base‘𝐵) & ⊢ 𝑌 = (Base‘𝐷) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐸) ⇒ ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝑥 ∈ 𝑋 ↦ ((𝐾‘𝑥)(⟨((1st ‘𝑀)‘𝑥), ((1st ‘𝑃)‘𝑥)⟩ · ((1st ‘𝑅)‘𝑥))(𝐹‘𝑥))), (𝑦 ∈ 𝑌 ↦ ((𝐿‘𝑦)(⟨((1st ‘𝑁)‘𝑦), ((1st ‘𝑄)‘𝑦)⟩ ∙ ((1st ‘𝑆)‘𝑦))(𝐺‘𝑦)))⟩) | ||
| Syntax | cswapf 49241 | Extend class notation with the class of swap functors. |
| class swapF | ||
| Definition | df-swapf 49242* |
Define the swap functor from (𝐶 ×c 𝐷) to (𝐷
×c 𝐶) by
swapping all objects (swapf1 49254) and morphisms (swapf2 49256) .
Such functor is called a "swap functor" in https://arxiv.org/pdf/2302.07810 49256 or a "twist functor" in https://arxiv.org/pdf/2508.01886 49256, the latter of which finds its counterpart as "twisting map" in https://arxiv.org/pdf/2411.04102 49256 for tensor product of algebras. The "swap functor" or "twisting map" is often denoted as a small tau 𝜏 in literature. However, the term "twist functor" is defined differently in https://arxiv.org/pdf/1208.4046 49256 and thus not adopted here. tpos I depends on more mathbox theorems, and thus are not adopted here. See dfswapf2 49243 for an alternate definition. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(𝑥 ∈ 𝑏 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (𝑓 ∈ (𝑢ℎ𝑣) ↦ ∪ ◡{𝑓}))⟩) | ||
| Theorem | dfswapf2 49243* | Alternate definition of swapF (df-swapf 49242). (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ swapF = (𝑐 ∈ V, 𝑑 ∈ V ↦ ⦋(𝑐 ×c 𝑑) / 𝑠⦌⦋(Base‘𝑠) / 𝑏⦌⦋(Hom ‘𝑠) / ℎ⦌⟨(tpos I ↾ 𝑏), (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (tpos I ↾ (𝑢ℎ𝑣)))⟩) | ||
| Theorem | swapfval 49244* | Value of the swap functor. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) ⇒ ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩) | ||
| Theorem | swapfelvv 49245 | A swap functor is an ordered pair. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V)) | ||
| Theorem | swapf2fvala 49246* | The morphism part of the swap functor. See also swapf2fval 49247. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) ⇒ ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))) | ||
| Theorem | swapf2fval 49247* | The morphism part of the swap functor. See also swapf2fvala 49246. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))) | ||
| Theorem | swapf1vala 49248* | The object part of the swap functor. See also swapf1val 49249. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥})) | ||
| Theorem | swapf1val 49249* | The object part of the swap functor. See also swapf1vala 49248. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑂 = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥})) | ||
| Theorem | swapf2fn 49250 | The morphism part of the swap functor is a function on the Cartesian square of the base set. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑃 Fn (𝐵 × 𝐵)) | ||
| Theorem | swapf1a 49251 | The object part of the swap functor swaps the objects. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) | ||
| Theorem | swapf2vala 49252* | The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) ⇒ ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓})) | ||
| Theorem | swapf2a 49253 | The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩) | ||
| Theorem | swapf1 49254 | The object part of the swap functor swaps the objects. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) ⇒ ⊢ (𝜑 → (𝑋𝑂𝑌) = ⟨𝑌, 𝑋⟩) | ||
| Theorem | swapf2val 49255* | The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷)) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆)) ⇒ ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩) = (𝑓 ∈ (⟨𝑋, 𝑌⟩𝐻⟨𝑍, 𝑊⟩) ↦ ∪ ◡{𝑓})) | ||
| Theorem | swapf2 49256 | The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷)) & ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑍)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑊)) ⇒ ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩)𝐺) = ⟨𝐺, 𝐹⟩) | ||
| Theorem | swapf1f1o 49257 | The object part of the swap functor is a bijection between base sets. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (Base‘𝑇) ⇒ ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→𝐴) | ||
| Theorem | swapf2f1o 49258 | The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐻 = (Hom ‘𝑆) & ⊢ 𝐽 = (Hom ‘𝑇) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐷)) ⇒ ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑃⟨𝑍, 𝑊⟩):(⟨𝑋, 𝑌⟩𝐻⟨𝑍, 𝑊⟩)–1-1-onto→(⟨𝑌, 𝑋⟩𝐽⟨𝑊, 𝑍⟩)) | ||
| Theorem | swapf2f1oa 49259 | The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐻 = (Hom ‘𝑆) & ⊢ 𝐽 = (Hom ‘𝑇) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌))) | ||
| Theorem | swapf2f1oaALT 49260 | Alternate proof of swapf2f1oa 49259. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐻 = (Hom ‘𝑆) & ⊢ 𝐽 = (Hom ‘𝑇) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌))) | ||
| Theorem | swapfid 49261 | Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfida 49262. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷)) & ⊢ 1 = (Id‘𝑆) & ⊢ 𝐼 = (Id‘𝑇) ⇒ ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩𝑃⟨𝑋, 𝑌⟩)‘( 1 ‘⟨𝑋, 𝑌⟩)) = (𝐼‘(𝑂‘⟨𝑋, 𝑌⟩))) | ||
| Theorem | swapfida 49262 | Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 49261. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 1 = (Id‘𝑆) & ⊢ 𝐼 = (Id‘𝑇) ⇒ ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝑂‘𝑋))) | ||
| Theorem | swapfcoa 49263 | Composition in the source category is mapped to composition in the target. (𝜑 → 𝐶 ∈ Cat) and (𝜑 → 𝐷 ∈ Cat) can be replaced by a weaker hypothesis (𝜑 → 𝑆 ∈ Cat). (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝑆) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍)) & ⊢ · = (comp‘𝑆) & ⊢ ∙ = (comp‘𝑇) ⇒ ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝑃𝑍)‘𝑁)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝑀))) | ||
| Theorem | swapffunc 49264 | The swap functor is a functor. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃) | ||
| Theorem | swapfffth 49265 | The swap functor is a fully faithful functor. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩) ⇒ ⊢ (𝜑 → 𝑂((𝑆 Full 𝑇) ∩ (𝑆 Faith 𝑇))𝑃) | ||
| Theorem | swapffunca 49266 | The swap functor is a functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) ⇒ ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (𝑆 Func 𝑇)) | ||
| Theorem | swapfiso 49267 | The swap functor is an isomorphism between product categories. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐸 = (CatCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑈) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) & ⊢ 𝐼 = (Iso‘𝐸) ⇒ ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (𝑆𝐼𝑇)) | ||
| Theorem | swapciso 49268 | The product category is categorically isomorphic to the swapped product category. (Contributed by Zhi Wang, 8-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝑆 = (𝐶 ×c 𝐷) & ⊢ 𝑇 = (𝐷 ×c 𝐶) & ⊢ 𝐸 = (CatCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑈) & ⊢ (𝜑 → 𝑇 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑆( ≃𝑐 ‘𝐸)𝑇) | ||
| Theorem | oppc1stflem 49269* | A utility theorem for proving theorems on projection functors of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶𝐹𝐷)) = (𝑂𝐹𝑃)) & ⊢ 𝐹 = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ 𝑌) ⇒ ⊢ (𝜑 → ( oppFunc ‘(𝐶𝐹𝐷)) = (𝑂𝐹𝑃)) | ||
| Theorem | oppc1stf 49270 | The opposite functor of the first projection functor is the first projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( oppFunc ‘(𝐶 1stF 𝐷)) = (𝑂 1stF 𝑃)) | ||
| Theorem | oppc2ndf 49271 | The opposite functor of the second projection functor is the second projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( oppFunc ‘(𝐶 2ndF 𝐷)) = (𝑂 2ndF 𝑃)) | ||
| Theorem | 1stfpropd 49272 | If two categories have the same set of objects, morphisms, and compositions, then they have same first projection functors. (Contributed by Zhi Wang, 20-Nov-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) & ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐴 ∈ Cat) & ⊢ (𝜑 → 𝐵 ∈ Cat) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (𝐴 1stF 𝐶) = (𝐵 1stF 𝐷)) | ||
| Theorem | 2ndfpropd 49273 | If two categories have the same set of objects, morphisms, and compositions, then they have same second projection functors. (Contributed by Zhi Wang, 20-Nov-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) & ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐴 ∈ Cat) & ⊢ (𝜑 → 𝐵 ∈ Cat) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (𝐴 2ndF 𝐶) = (𝐵 2ndF 𝐷)) | ||
| Theorem | diagpropd 49274 | If two categories have the same set of objects, morphisms, and compositions, then they have same diagonal functors. (Contributed by Zhi Wang, 20-Nov-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) & ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) & ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) & ⊢ (𝜑 → 𝐴 ∈ Cat) & ⊢ (𝜑 → 𝐵 ∈ Cat) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (𝐴Δfunc𝐶) = (𝐵Δfunc𝐷)) | ||
| Theorem | cofuswapfcl 49275 | The bifunctor pre-composed with a swap functor is a bifunctor. (Contributed by Zhi Wang, 10-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷))) ⇒ ⊢ (𝜑 → 𝐺 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) | ||
| Theorem | cofuswapf1 49276 | The object part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = (𝑌(1st ‘𝐹)𝑋)) | ||
| Theorem | cofuswapf2 49277 | The morphism part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑍)) & ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐽𝑊)) ⇒ ⊢ (𝜑 → (𝑀(⟨𝑋, 𝑌⟩(2nd ‘𝐺)⟨𝑍, 𝑊⟩)𝑁) = (𝑁(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑊, 𝑍⟩)𝑀)) | ||
| Theorem | tposcurf1cl 49278 | The partially evaluated transposed curry functor is a functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋)) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸)) | ||
| Theorem | tposcurf11 49279 | Value of the double evaluated transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = (𝑌(1st ‘𝐹)𝑋)) | ||
| Theorem | tposcurf12 49280 | The partially evaluated transposed curry functor at a morphism. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍)) ⇒ ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (𝐻(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑋⟩)( 1 ‘𝑋))) | ||
| Theorem | tposcurf1 49281* | Value of the object part of the transposed curry functor. (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ (𝑦(1st ‘𝐹)𝑋)), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦𝐽𝑧) ↦ (𝑔(⟨𝑦, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑋⟩)( 1 ‘𝑋))))⟩) | ||
| Theorem | tposcurf2 49282* | Value of the transposed curry functor at a morphism. (Contributed by Zhi Wang, 10-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) ⇒ ⊢ (𝜑 → 𝐿 = (𝑧 ∈ 𝐵 ↦ ((𝐼‘𝑧)(⟨𝑧, 𝑋⟩(2nd ‘𝐹)⟨𝑧, 𝑌⟩)𝐾))) | ||
| Theorem | tposcurf2val 49283 | Value of a component of the transposed curry functor natural transformation. (Contributed by Zhi Wang, 10-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐿‘𝑍) = ((𝐼‘𝑍)(⟨𝑍, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑌⟩)𝐾)) | ||
| Theorem | tposcurf2cl 49284 | The transposed curry functor at a morphism is a natural transformation. (Contributed by Zhi Wang, 10-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Id‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐿 = ((𝑋(2nd ‘𝐺)𝑌)‘𝐾)) & ⊢ 𝑁 = (𝐷 Nat 𝐸) ⇒ ⊢ (𝜑 → 𝐿 ∈ (((1st ‘𝐺)‘𝑋)𝑁((1st ‘𝐺)‘𝑌))) | ||
| Theorem | tposcurfcl 49285 | The transposed curry functor of a functor 𝐹:𝐷 × 𝐶⟶𝐸 is a functor tposcurry (𝐹):𝐶⟶(𝐷⟶𝐸). (Contributed by Zhi Wang, 9-Oct-2025.) |
| ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))) & ⊢ 𝑄 = (𝐷 FuncCat 𝐸) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝑄)) | ||
| Theorem | diag1 49286* | The constant functor of 𝑋. Example 3.20(2) of [Adamek] p. 30. (Contributed by Zhi Wang, 17-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ 𝑋), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋)))⟩) | ||
| Theorem | diag1a 49287* | The constant functor of 𝑋. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 1 = (Id‘𝐶) ⇒ ⊢ (𝜑 → 𝐾 = ⟨(𝐵 × {𝑋}), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ ((𝑦𝐽𝑧) × {( 1 ‘𝑋)}))⟩) | ||
| Theorem | diag1f1lem 49288 | The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. Note that (𝜑 → (𝑀 = 𝑁 ↔ 𝑋 = 𝑌)) also holds because of diag1f1 49289 and f1fveq 7219. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑀 = ((1st ‘𝐿)‘𝑋) & ⊢ 𝑁 = ((1st ‘𝐿)‘𝑌) ⇒ ⊢ (𝜑 → (𝑀 = 𝑁 → 𝑋 = 𝑌)) | ||
| Theorem | diag1f1 49289 | The object part of the diagonal functor is 1-1 if 𝐵 is non-empty. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐵 ≠ ∅) ⇒ ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶)) | ||
| Theorem | diag2f1lem 49290 | Lemma for diag2f1 49291. The converse is trivial (fveq2 6840). (Contributed by Zhi Wang, 21-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = ((𝑋(2nd ‘𝐿)𝑌)‘𝐺) → 𝐹 = 𝐺)) | ||
| Theorem | diag2f1 49291 | If 𝐵 is non-empty, the morphism part of a diagonal functor is injective functions from hom-sets into sets of natural transformations. (Contributed by Zhi Wang, 21-Oct-2025.) |
| ⊢ 𝐿 = (𝐶Δfunc𝐷) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ 𝑁 = (𝐷 Nat 𝐶) ⇒ ⊢ (𝜑 → (𝑋(2nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌))) | ||
| Theorem | fucofulem1 49292 | Lemma for proving functor theorems. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) & ⊢ ((𝜑 ∧ (𝜃 ∧ 𝜏)) → 𝜂) & ⊢ 𝜒 & ⊢ ((𝜑 ∧ 𝜂) → 𝜃) & ⊢ ((𝜑 ∧ 𝜂) → 𝜏) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜂)) | ||
| Theorem | fucofulem2 49293* | Lemma for proving functor theorems. Maybe consider eufnfv 7185 to prove the uniqueness of a functor. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ 𝐵 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) & ⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))) ⇒ ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))) | ||
| Theorem | fuco2el 49294 | Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺)) | ||
| Theorem | fuco2eld 49295 | Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅)) & ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩) & ⊢ (𝜑 → 𝐾𝑆𝐿) & ⊢ (𝜑 → 𝐹𝑅𝐺) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝑊) | ||
| Theorem | fuco2eld2 49296 | Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅)) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ Rel 𝑆 & ⊢ Rel 𝑅 ⇒ ⊢ (𝜑 → 𝑈 = ⟨⟨(1st ‘(1st ‘𝑈)), (2nd ‘(1st ‘𝑈))⟩, ⟨(1st ‘(2nd ‘𝑈)), (2nd ‘(2nd ‘𝑈))⟩⟩) | ||
| Theorem | fuco2eld3 49297 | Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅)) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ Rel 𝑆 & ⊢ Rel 𝑅 ⇒ ⊢ (𝜑 → ((1st ‘(1st ‘𝑈))𝑆(2nd ‘(1st ‘𝑈)) ∧ (1st ‘(2nd ‘𝑈))𝑅(2nd ‘(2nd ‘𝑈)))) | ||
| Syntax | cfuco 49298 | Extend class notation with functor composition bifunctors. |
| class ∘F | ||
| Definition | df-fuco 49299* | Definition of functor composition bifunctors. Given three categories 𝐶, 𝐷, and 𝐸, (⟨𝐶, 𝐷⟩ ∘F 𝐸) is a functor from the product category of two categories of functors to a category of functors (fucofunc 49341). The object part maps two functors to their composition (fuco11 49308 and fuco11b 49319). The morphism part defines the "composition" of two natural transformations (fuco22 49321) into another natural transformation (fuco22nat 49328) such that a "cube-like" diagram commutes. The naturality property also gives an alternate definition (fuco23a 49334). Note that such "composition" is different from fucco 17907 because they "compose" along different "axes". (Contributed by Zhi Wang, 29-Sep-2025.) |
| ⊢ ∘F = (𝑝 ∈ V, 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌⦋((𝑑 Func 𝑒) × (𝑐 Func 𝑑)) / 𝑤⦌⟨( ∘func ↾ 𝑤), (𝑢 ∈ 𝑤, 𝑣 ∈ 𝑤 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝑑 Nat 𝑒)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝑐 Nat 𝑑)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝑐) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝑒)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩) | ||
| Theorem | fucofvalg 49300* | Value of the function giving the functor composition bifunctor. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → 𝑃 ∈ 𝑈) & ⊢ (𝜑 → (1st ‘𝑃) = 𝐶) & ⊢ (𝜑 → (2nd ‘𝑃) = 𝐷) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → (𝑃 ∘F 𝐸) = ⚬ ) & ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))) ⇒ ⊢ (𝜑 → ⚬ = ⟨( ∘func ↾ 𝑊), (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ ⦋(1st ‘(2nd ‘𝑢)) / 𝑓⦌⦋(1st ‘(1st ‘𝑢)) / 𝑘⦌⦋(2nd ‘(1st ‘𝑢)) / 𝑙⦌⦋(1st ‘(2nd ‘𝑣)) / 𝑚⦌⦋(1st ‘(1st ‘𝑣)) / 𝑟⦌(𝑏 ∈ ((1st ‘𝑢)(𝐷 Nat 𝐸)(1st ‘𝑣)), 𝑎 ∈ ((2nd ‘𝑢)(𝐶 Nat 𝐷)(2nd ‘𝑣)) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑚‘𝑥))(⟨(𝑘‘(𝑓‘𝑥)), (𝑘‘(𝑚‘𝑥))⟩(comp‘𝐸)(𝑟‘(𝑚‘𝑥)))(((𝑓‘𝑥)𝑙(𝑚‘𝑥))‘(𝑎‘𝑥))))))⟩) | ||
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