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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | up1st2nd2 50101 | Rewrite the universal property predicate with separated parts. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝜑 → 𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊)) ⇒ ⊢ (𝜑 → (1st ‘𝑋)(𝐹(𝐷 UP 𝐸)𝑊)(2nd ‘𝑋)) | ||
| Theorem | uprcl2 50102 | Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀) ⇒ ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | ||
| Theorem | uprcl3 50103 | Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀) & ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝜑 → 𝑊 ∈ 𝐶) | ||
| Theorem | uprcl4 50104 | Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀) & ⊢ 𝐵 = (Base‘𝐷) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐵) | ||
| Theorem | uprcl5 50105 | Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀) & ⊢ 𝐽 = (Hom ‘𝐸) ⇒ ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) | ||
| Theorem | uobrcl 50106 | Reverse closure for universal object. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ (𝑋 ∈ dom (𝐹(𝐷 UP 𝐸)𝑊) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | ||
| Theorem | isup2 50107* | The universal property of a universal pair. (Contributed by Zhi Wang, 24-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀) ⇒ ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀)) | ||
| Theorem | upeu3 50108* | The universal pair ⟨𝑋, 𝑀⟩ from object 𝑊 to functor ⟨𝐹, 𝐺⟩ is essentially unique (strong form) if it exists. (Contributed by Zhi Wang, 24-Sep-2025.) |
| ⊢ (𝜑 → 𝐼 = (Iso‘𝐷)) & ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))) & ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀) & ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁) ⇒ ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋𝐼𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟) ⚬ 𝑀)) | ||
| Theorem | upeu4 50109 | Generate a new universal morphism through an isomorphism from an existing universal object, and pair with the codomain of the isomorphism to form a universal pair. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ (𝜑 → 𝐼 = (Iso‘𝐷)) & ⊢ (𝜑 → ⚬ = (⟨𝑊, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑌))) & ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌)) & ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾) ⚬ 𝑀)) ⇒ ⊢ (𝜑 → 𝑌(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑁) | ||
| Theorem | uptposlem 50110 | Lemma for uptpos 50111. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ (𝜑 → tpos 𝐺 = 𝐻) ⇒ ⊢ (𝜑 → tpos 𝐻 = 𝐺) | ||
| Theorem | uptpos 50111 | Rewrite the predicate of universal property in the form of opposite functor. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ (𝜑 → tpos 𝐺 = 𝐻) ⇒ ⊢ (𝜑 → 𝑋(⟨𝐹, tpos 𝐻⟩(𝑂 UP 𝑃)𝑊)𝑀) | ||
| Theorem | oppcuprcl4 50112 | Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ 𝑂 = (oppCat‘𝐷) & ⊢ 𝐵 = (Base‘𝐷) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐵) | ||
| Theorem | oppcuprcl3 50113 | Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ 𝑃 = (oppCat‘𝐸) & ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝜑 → 𝑊 ∈ 𝐶) | ||
| Theorem | oppcuprcl5 50114 | Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ 𝑃 = (oppCat‘𝐸) & ⊢ 𝐽 = (Hom ‘𝐸) ⇒ ⊢ (𝜑 → 𝑀 ∈ ((𝐹‘𝑋)𝐽𝑊)) | ||
| Theorem | oppcuprcl2 50115 | Reverse closure for the class of universal property in opposite categories. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ 𝑃 = (oppCat‘𝐸) & ⊢ 𝑂 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → tpos 𝐺 = 𝐻) ⇒ ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐻) | ||
| Theorem | uprcl2a 50116 | Reverse closure for the class of universal property. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝑂 Func 𝑃)) | ||
| Theorem | oppfuprcl 50117 | Reverse closure for the class of universal property for opposite functors. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ 𝐺 = ( oppFunc ‘𝐹) & ⊢ 𝑂 = (oppCat‘𝐷) & ⊢ 𝑃 = (oppCat‘𝐸) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) | ||
| Theorem | oppfuprcl2 50118 | Reverse closure for the class of universal property for opposite functors. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ (𝜑 → 𝑋(𝐺(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ 𝐺 = ( oppFunc ‘𝐹) & ⊢ 𝑂 = (oppCat‘𝐷) & ⊢ 𝑃 = (oppCat‘𝐸) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 = ⟨𝐴, 𝐵⟩) ⇒ ⊢ (𝜑 → 𝐴(𝐷 Func 𝐸)𝐵) | ||
| Theorem | oppcup3lem 50119* | Lemma for oppcup3 50122. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ ((𝐹‘𝑦)𝐽𝑍)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑛 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩𝑂𝑍)((𝑦𝐺𝑋)‘𝑘))) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑍)) ⇒ ⊢ (𝜑 → ∃!𝑙 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩𝑂𝑍)((𝑌𝐺𝑋)‘𝑙))) | ||
| Theorem | oppcup 50120* | The universal pair ⟨𝑋, 𝑀⟩ from a functor to an object is universal from an object to a functor in the opposite category. (Contributed by Zhi Wang, 24-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ ∙ = (comp‘𝐸) & ⊢ (𝜑 → 𝑊 ∈ 𝐶) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ ((𝐹‘𝑋)𝐽𝑊)) & ⊢ 𝑂 = (oppCat‘𝐷) & ⊢ 𝑃 = (oppCat‘𝐸) ⇒ ⊢ (𝜑 → (𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘)))) | ||
| Theorem | oppcup2 50121* | The universal property for the universal pair ⟨𝑋, 𝑀⟩ from a functor to an object, expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ ∙ = (comp‘𝐸) & ⊢ 𝑂 = (oppCat‘𝐷) & ⊢ 𝑃 = (oppCat‘𝐸) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋(⟨𝐹, tpos 𝐺⟩(𝑂 UP 𝑃)𝑊)𝑀) ⇒ ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ ((𝐹‘𝑦)𝐽𝑊)∃!𝑘 ∈ (𝑦𝐻𝑋)𝑔 = (𝑀(⟨(𝐹‘𝑦), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑦𝐺𝑋)‘𝑘))) | ||
| Theorem | oppcup3 50122* | The universal property for the universal pair ⟨𝑋, 𝑀⟩ from a functor to an object, expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ ∙ = (comp‘𝐸) & ⊢ 𝑂 = (oppCat‘𝐷) & ⊢ 𝑃 = (oppCat‘𝐸) & ⊢ (𝜑 → 𝑋(⟨𝐹, 𝑇⟩(𝑂 UP 𝑃)𝑊)𝑀) & ⊢ (𝜑 → tpos 𝑇 = 𝐺) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ((𝐹‘𝑌)𝐽𝑊)) ⇒ ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑋)𝑁 = (𝑀(⟨(𝐹‘𝑌), (𝐹‘𝑋)⟩ ∙ 𝑊)((𝑌𝐺𝑋)‘𝑘))) | ||
| Theorem | uptrlem1 50123* | Lemma for uptr 50126. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ ∙ = (comp‘𝐷) & ⊢ ⚬ = (comp‘𝐸) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷)) & ⊢ (𝜑 → (𝑀‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝐹‘𝑍))) & ⊢ (𝜑 → ((𝑋𝑁(𝐹‘𝑍))‘𝐴) = 𝐵) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) & ⊢ (𝜑 → 𝑀((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑁) & ⊢ (𝜑 → (⟨𝑀, 𝑁⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩) ⇒ ⊢ (𝜑 → (∀ℎ ∈ (𝑌𝐽(𝐾‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)ℎ = (((𝑍𝐿𝑊)‘𝑘)(⟨𝑌, (𝐾‘𝑍)⟩ ⚬ (𝐾‘𝑊))𝐵) ↔ ∀𝑔 ∈ (𝑋𝐼(𝐹‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)𝑔 = (((𝑍𝐺𝑊)‘𝑘)(⟨𝑋, (𝐹‘𝑍)⟩ ∙ (𝐹‘𝑊))𝐴))) | ||
| Theorem | uptrlem2 50124* | Lemma for uptr 50126. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ ∙ = (comp‘𝐷) & ⊢ ⚬ = (comp‘𝐸) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ 𝐴) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼((1st ‘𝐹)‘𝑍))) & ⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) & ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺) ⇒ ⊢ (𝜑 → (∀ℎ ∈ (𝑌𝐽((1st ‘𝐺)‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)ℎ = (((𝑍(2nd ‘𝐺)𝑊)‘𝑘)(⟨𝑌, ((1st ‘𝐺)‘𝑍)⟩ ⚬ ((1st ‘𝐺)‘𝑊))𝑁) ↔ ∀𝑔 ∈ (𝑋𝐼((1st ‘𝐹)‘𝑊))∃!𝑘 ∈ (𝑍𝐻𝑊)𝑔 = (((𝑍(2nd ‘𝐹)𝑊)‘𝑘)(⟨𝑋, ((1st ‘𝐹)‘𝑍)⟩ ∙ ((1st ‘𝐹)‘𝑊))𝑀))) | ||
| Theorem | uptrlem3 50125 | Lemma for uptr 50126. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ (𝜑 → (𝑅‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆) & ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) & ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍))) & ⊢ 𝐴 = (Base‘𝐶) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁)) | ||
| Theorem | uptr 50126 | Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ (𝜑 → (𝑅‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆) & ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) & ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽(𝐹‘𝑍))) ⇒ ⊢ (𝜑 → (𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁)) | ||
| Theorem | uptri 50127 | Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ (𝜑 → (𝑅‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝑅((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))𝑆) & ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝐾, 𝐿⟩) & ⊢ (𝜑 → ((𝑋𝑆(𝐹‘𝑍))‘𝑀) = 𝑁) & ⊢ (𝜑 → 𝑍(⟨𝐹, 𝐺⟩(𝐶 UP 𝐷)𝑋)𝑀) ⇒ ⊢ (𝜑 → 𝑍(⟨𝐾, 𝐿⟩(𝐶 UP 𝐸)𝑌)𝑁) | ||
| Theorem | uptra 50128 | Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) & ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐽((1st ‘𝐹)‘𝑍))) ⇒ ⊢ (𝜑 → (𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀 ↔ 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁)) | ||
| Theorem | uptrar 50129 | Universal property and fully faithful functor. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) & ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → (◡(𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑁) = 𝑀) & ⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) ⇒ ⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) | ||
| Theorem | uptrai 50130 | Universal property and fully faithful functor. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) & ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺) & ⊢ (𝜑 → ((𝑋(2nd ‘𝐾)((1st ‘𝐹)‘𝑍))‘𝑀) = 𝑁) & ⊢ (𝜑 → 𝑍(𝐹(𝐶 UP 𝐷)𝑋)𝑀) ⇒ ⊢ (𝜑 → 𝑍(𝐺(𝐶 UP 𝐸)𝑌)𝑁) | ||
| Theorem | uobffth 50131 | A fully faithful functor generates equal sets of universal objects. (Contributed by Zhi Wang, 19-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺) & ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌) & ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸))) ⇒ ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌)) | ||
| Theorem | uobeqw 50132 | 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 50133 | 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 50134 | 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 50135 | 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 50136* | 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 50137 | 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 50138 | 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 50139 | 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 50140 | 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 50141 | 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 50142 | 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 50143 | 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 50144 | 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 50145 | 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 50146 | 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 50147 | 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 50148 | Initial objects are terminal in the opposite category. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ (TermO‘(oppCat‘𝐶))) | ||
| Theorem | oppctermo 50149 | 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 50150 | 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 50151 | 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 50152 | Lemma for initopropdlem 50153, termopropdlem 50154, and zeroopropdlem 50155. (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 50153 | Lemma for initopropd 50156. (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 50154 | Lemma for termopropd 50157. (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 50155 | Lemma for zeroopropd 50158. (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 50156 | 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 50157 | 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 50158 | 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 50159 | The binary product of categories is a proper operator, so it can be used with ovprc1 7455, elbasov 17312, strov2rcl 17313, and so on. See reldmxpcALT 50160 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 50160 | Alternate proof of reldmxpc 50159. (Contributed by Zhi Wang, 15-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Rel dom ×c | ||
| Theorem | elxpcbasex1 50161 | 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 50162 | Alternate proof of elxpcbasex1 50161. (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 50163 | 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 50164 | Alternate proof of elxpcbasex2 50163. (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 50165 | 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 50166* | 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 50167 | 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 50168 | 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 50169 | 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 50170 | 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 50171* | 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 50172 | Extend class notation with the class of swap functors. |
| class swapF | ||
| Definition | df-swapf 50173* |
Define the swap functor from (𝐶 ×c 𝐷) to (𝐷
×c 𝐶) by
swapping all objects (swapf1 50185) and morphisms (swapf2 50187) .
Such functor is called a "swap functor" in https://arxiv.org/pdf/2302.07810 50187 or a "twist functor" in https://arxiv.org/pdf/2508.01886 50187, the latter of which finds its counterpart as "twisting map" in https://arxiv.org/pdf/2411.04102 50187 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 50187 and thus not adopted here. tpos I depends on more mathbox theorems, and thus are not adopted here. See dfswapf2 50174 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 50174* | Alternate definition of swapF (df-swapf 50173). (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 50175* | 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 50176 | 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 50177* | The morphism part of the swap functor. See also swapf2fval 50178. (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 50178* | The morphism part of the swap functor. See also swapf2fvala 50177. (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 50179* | The object part of the swap functor. See also swapf1val 50180. (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 50180* | The object part of the swap functor. See also swapf1vala 50179. (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 50181 | 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 50182 | 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 50183* | 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 50184 | 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 50185 | 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 50186* | 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 50187 | 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 50188 | 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 50189 | 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 50190 | 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 50191 | Alternate proof of swapf2f1oa 50190. (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 50192 | Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfida 50193. (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 50193 | Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 50192. (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 50194 | 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 50195 | 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 50196 | 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 50197 | 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 50198 | 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 50199 | 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 50200* | 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)) | ||
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