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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | upeu3 50001* | 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 50002 | 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 50003 | Lemma for uptpos 50004. (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 50004 | 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 50005 | 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 50006 | 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 50007 | 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 50008 | 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 50009 | 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 50010 | 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 50011 | 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 50012* | Lemma for oppcup3 50015. (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 50013* | 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 50014* | 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 50015* | 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 50016* | Lemma for uptr 50019. (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 50017* | Lemma for uptr 50019. (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 50018 | Lemma for uptr 50019. (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 50019 | 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 50020 | 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 50021 | 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 50022 | 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 50023 | 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 50024 | 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 50025 | 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 50026 | 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 50027 | 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 50028 | 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 50029* | 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 50030 | 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 50031 | 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 50032 | 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 50033 | 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 50034 | 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 50035 | 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 50036 | 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 50037 | 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 50038 | 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 50039 | 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 50040 | 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 50041 | Initial objects are terminal in the opposite category. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ (TermO‘(oppCat‘𝐶))) | ||
| Theorem | oppctermo 50042 | 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 50043 | 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 50044 | 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 50045 | Lemma for initopropdlem 50046, termopropdlem 50047, and zeroopropdlem 50048. (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 50046 | Lemma for initopropd 50049. (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 50047 | Lemma for termopropd 50050. (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 50048 | Lemma for zeroopropd 50051. (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 50049 | 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 50050 | 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 50051 | 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 50052 | The binary product of categories is a proper operator, so it can be used with ovprc1 7451, elbasov 17282, strov2rcl 17283, and so on. See reldmxpcALT 50053 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 50053 | Alternate proof of reldmxpc 50052. (Contributed by Zhi Wang, 15-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Rel dom ×c | ||
| Theorem | elxpcbasex1 50054 | 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 50055 | Alternate proof of elxpcbasex1 50054. (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 50056 | 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 50057 | Alternate proof of elxpcbasex2 50056. (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 50058 | 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 50059* | 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 50060 | 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 50061 | 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 50062 | 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 50063 | 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 50064* | 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 50065 | Extend class notation with the class of swap functors. |
| class swapF | ||
| Definition | df-swapf 50066* |
Define the swap functor from (𝐶 ×c 𝐷) to (𝐷
×c 𝐶) by
swapping all objects (swapf1 50078) and morphisms (swapf2 50080) .
Such functor is called a "swap functor" in https://arxiv.org/pdf/2302.07810 50080 or a "twist functor" in https://arxiv.org/pdf/2508.01886 50080, the latter of which finds its counterpart as "twisting map" in https://arxiv.org/pdf/2411.04102 50080 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 50080 and thus not adopted here. tpos I depends on more mathbox theorems, and thus are not adopted here. See dfswapf2 50067 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 50067* | Alternate definition of swapF (df-swapf 50066). (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 50068* | 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 50069 | 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 50070* | The morphism part of the swap functor. See also swapf2fval 50071. (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 50071* | The morphism part of the swap functor. See also swapf2fvala 50070. (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 50072* | The object part of the swap functor. See also swapf1val 50073. (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 50073* | The object part of the swap functor. See also swapf1vala 50072. (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 50074 | 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 50075 | 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 50076* | 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 50077 | 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 50078 | 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 50079* | 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 50080 | 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 50081 | 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 50082 | 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 50083 | 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 50084 | Alternate proof of swapf2f1oa 50083. (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 50085 | Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfida 50086. (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 50086 | Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 50085. (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 50087 | 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 50088 | 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 50089 | 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 50090 | 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 50091 | 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 50092 | 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 50093* | 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 50094 | 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 50095 | 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 50096 | 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 50097 | 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 50098 | 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 50099 | 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 50100 | 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)) | ||
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