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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | 2oppffunc 49401 | The opposite functor of an opposite functor is a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025.) The functor in opposite categories does not have to be an opposite functor. (Revised by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑃)) ⇒ ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝐶 Func 𝐷)) | ||
| Theorem | funcoppc3 49402 | A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 4-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → 𝐹(𝑂 Func 𝑃)tpos 𝐺) & ⊢ (𝜑 → 𝐺 Fn (𝐴 × 𝐵)) ⇒ ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) | ||
| Theorem | oppff1 49403 | The operation generating opposite functors is injective. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) ⇒ ⊢ ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1→(𝑂 Func 𝑃) | ||
| Theorem | oppff1o 49404 | The operation generating opposite functors is bijective. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( oppFunc ↾ (𝐶 Func 𝐷)):(𝐶 Func 𝐷)–1-1-onto→(𝑂 Func 𝑃)) | ||
| Theorem | cofuoppf 49405 | Composition of opposite functors. (Contributed by Zhi Wang, 26-Nov-2025.) |
| ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐾) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸)) ⇒ ⊢ (𝜑 → (( oppFunc ‘𝐺) ∘func ( oppFunc ‘𝐹)) = ( oppFunc ‘𝐾)) | ||
| Theorem | imasubc 49406* | An image of a full functor is a full subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ 𝑆 = (𝐹 “ 𝐴) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝))) & ⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐽 = (Homf ‘𝐸) ⇒ ⊢ (𝜑 → (𝐾 Fn (𝑆 × 𝑆) ∧ 𝑆 ⊆ 𝐶 ∧ (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾)) | ||
| Theorem | imasubc2 49407* | An image of a full functor is a (full) subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ 𝑆 = (𝐹 “ 𝐴) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝))) & ⊢ (𝜑 → 𝐹(𝐷 Full 𝐸)𝐺) ⇒ ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸)) | ||
| Theorem | imassc 49408* | An image of a functor satisfies the subcategory subset relation. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ 𝑆 = (𝐹 “ 𝐴) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝))) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ 𝐽 = (Homf ‘𝐸) ⇒ ⊢ (𝜑 → 𝐾 ⊆cat 𝐽) | ||
| Theorem | imaid 49409* | An image of a functor preserves the identity morphism. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ 𝑆 = (𝐹 “ 𝐴) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝))) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ 𝐼 = (Id‘𝐸) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝑋𝐾𝑋)) | ||
| Theorem | imaf1co 49410* | An image of a functor whose object part is injective preserves the composition. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ 𝑆 = (𝐹 “ 𝐴) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝))) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ ∙ = (comp‘𝐸) & ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → 𝑍 ∈ 𝑆) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐾𝑌)) & ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐾𝑍)) ⇒ ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝑀) ∈ (𝑋𝐾𝑍)) | ||
| Theorem | imasubc3 49411* | An image of a functor injective on objects is a subcategory. Remark 4.2(3) of [Adamek] p. 48. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ 𝑆 = (𝐹 “ 𝐴) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝))) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → Fun ◡𝐹) ⇒ ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐸)) | ||
| Theorem | fthcomf 49412* | Source categories of a faithful functor have the same base, hom-sets and composition operation if the composition is compatible in images of the functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ (𝜑 → 𝐹(𝐴 Faith 𝐶)𝐺) & ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺) & ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴) ∧ 𝑧 ∈ (Base‘𝐴))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐴)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐴)𝑧))) → (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐶)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓)) = (((𝑦𝐺𝑧)‘𝑔)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐷)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑓))) ⇒ ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵)) | ||
| Theorem | idfth 49413 | The inclusion functor is a faithful functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) ⇒ ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐼 ∈ (𝐷 Faith 𝐸)) | ||
| Theorem | idemb 49414 | The inclusion functor is an embedding. Remark 4.4(1) in [Adamek] p. 49. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) ⇒ ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → (𝐼 ∈ (𝐷 Faith 𝐸) ∧ Fun ◡(1st ‘𝐼))) | ||
| Theorem | idsubc 49415 | The source category of an inclusion functor is a subcategory of the target category. See also Remark 4.4 in [Adamek] p. 49. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐷) ⇒ ⊢ (𝐼 ∈ (𝐷 Func 𝐸) → 𝐻 ∈ (Subcat‘𝐸)) | ||
| Theorem | idfullsubc 49416 | The source category of an inclusion functor is a full subcategory of the target category if the inclusion functor is full. Remark 4.4(2) in [Adamek] p. 49. See also ressffth 17864. (Contributed by Zhi Wang, 11-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ 𝐻 = (Homf ‘𝐷) & ⊢ 𝐽 = (Homf ‘𝐸) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝐼 ∈ (𝐷 Full 𝐸) → (𝐵 ⊆ 𝐶 ∧ (𝐽 ↾ (𝐵 × 𝐵)) = 𝐻)) | ||
| Theorem | cofidfth 49417 | If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then 𝐹 is faithful. Combined with cofidf1 49376, this theorem proves that 𝐹 is an embedding (a faithful functor injective on objects, remark 3.28(1) of [Adamek] p. 34). (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿) & ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼) ⇒ ⊢ (𝜑 → 𝐹(𝐷 Faith 𝐸)𝐺) | ||
| Theorem | fulloppf 49418 | The opposite functor of a full functor is also full. Proposition 3.43(d) in [Adamek] p. 39. (Contributed by Zhi Wang, 26-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Full 𝐷)) ⇒ ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Full 𝑃)) | ||
| Theorem | fthoppf 49419 | The opposite functor of a faithful functor is also faithful. Proposition 3.43(c) in [Adamek] p. 39. (Contributed by Zhi Wang, 26-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Faith 𝐷)) ⇒ ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Faith 𝑃)) | ||
| Theorem | ffthoppf 49420 | The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Zhi Wang, 26-Nov-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝑃 = (oppCat‘𝐷) & ⊢ (𝜑 → 𝐹 ∈ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))) ⇒ ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ ((𝑂 Full 𝑃) ∩ (𝑂 Faith 𝑃))) | ||
| Theorem | upciclem1 49421* | Lemma for upcic 49425, upeu 49426, and upeu2 49427. (Contributed by Zhi Wang, 16-Sep-2025.) (Proof shortened by Zhi Wang, 5-Nov-2025.) |
| ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑛 ∈ (𝑍𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑛 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀)) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌))) ⇒ ⊢ (𝜑 → ∃!𝑙 ∈ (𝑋𝐻𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑙)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)) | ||
| Theorem | upciclem2 49422 | Lemma for upciclem3 49423 and upeu2 49427. (Contributed by Zhi Wang, 19-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝐶) & ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) & ⊢ · = (comp‘𝐷) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)) ⇒ ⊢ (𝜑 → (((𝑋𝐺𝑍)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑍)𝐾))(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑍))𝑀) = (((𝑌𝐺𝑍)‘𝐿)(⟨𝑊, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑍))𝑁)) | ||
| Theorem | upciclem3 49423* | Lemma for upciclem4 49424. (Contributed by Zhi Wang, 17-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐶) & ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋))) & ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀)) & ⊢ · = (comp‘𝐷) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑋)) & ⊢ (𝜑 → 𝑀 = (((𝑌𝐺𝑋)‘𝐿)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁)) & ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)) ⇒ ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) = ((Id‘𝐷)‘𝑋)) | ||
| Theorem | upciclem4 49424* | Lemma for upcic 49425 and upeu 49426. (Contributed by Zhi Wang, 19-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐶) & ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋))) & ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀)) & ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌))) & ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁)) ⇒ ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐷)𝑌 ∧ ∃𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))) | ||
| Theorem | upcic 49425* | A universal property defines an object up to isomorphism given its existence. (Contributed by Zhi Wang, 17-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐶) & ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋))) & ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀)) & ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌))) & ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁)) ⇒ ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐷)𝑌) | ||
| Theorem | upeu 49426* | A universal property defines an essentially unique (strong form) pair of object 𝑋 and morphism 𝑀 if it exists. (Contributed by Zhi Wang, 19-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐶) & ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋))) & ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀)) & ⊢ (𝜑 → 𝑁 ∈ (𝑍𝐽(𝐹‘𝑌))) & ⊢ (𝜑 → ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁)) ⇒ ⊢ (𝜑 → ∃!𝑟 ∈ (𝑋(Iso‘𝐷)𝑌)𝑁 = (((𝑋𝐺𝑌)‘𝑟)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)) | ||
| Theorem | upeu2 49427* | Generate new universal morphism through isomorphism from existing universal object. (Contributed by Zhi Wang, 20-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐶) & ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋))) & ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀)) & ⊢ 𝐼 = (Iso‘𝐷) & ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐼𝑌)) & ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀)) ⇒ ⊢ (𝜑 → (𝑁 ∈ (𝑍𝐽(𝐹‘𝑌)) ∧ ∀𝑣 ∈ 𝐵 ∀𝑔 ∈ (𝑍𝐽(𝐹‘𝑣))∃!𝑙 ∈ (𝑌𝐻𝑣)𝑔 = (((𝑌𝐺𝑣)‘𝑙)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑣))𝑁))) | ||
| Syntax | cup 49428 | Extend class notation with the class of universal properties. |
| class UP | ||
| Definition | df-up 49429* |
Definition of the class of universal properties.
Given categories 𝐷 and 𝐸, if 𝐹:𝐷⟶𝐸 is a functor and 𝑊 an object of 𝐸, a universal pair from 𝑊 to 𝐹 is a pair ⟨𝑋, 𝑀⟩ consisting of an object 𝑋 of 𝐷 and a morphism 𝑀:𝑊⟶𝐹𝑋 of 𝐸, such that to every pair ⟨𝑦, 𝑔⟩ with 𝑦 an object of 𝐷 and 𝑔:𝑊⟶𝐹𝑦 a morphism of 𝐸, there is a unique morphism 𝑘:𝑋⟶𝑦 of 𝐷 with 𝐹𝑘 ⚬ 𝑀 = 𝑔. Such property is commonly referred to as a universal property. In our definition, it is denoted as 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀. Note that the universal pair is termed differently as "universal arrow" in p. 55 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf (retrieved 6 Oct 2025). Interestingly, the "universal arrow" is referring to the morphism 𝑀 instead of the pair near the end of the same piece of the text, causing name collision. The name "universal arrow" is also adopted in papers such as https://arxiv.org/pdf/2212.08981. Alternatively, the universal pair is called the "universal morphism" in Wikipedia (https://en.wikipedia.org/wiki/Universal_property) as well as published works, e.g., https://arxiv.org/pdf/2412.12179. But the pair ⟨𝑋, 𝑀⟩ should be named differently as the morphism 𝑀, and thus we call 𝑋 the universal object, 𝑀 the universal morphism, and ⟨𝑋, 𝑀⟩ the universal pair. Given its existence, such universal pair is essentially unique (upeu3 49450), and can be generated from an existing universal pair by isomorphisms (upeu4 49451). See also oppcup 49462 for the dual concept. (Contributed by Zhi Wang, 24-Sep-2025.) |
| ⊢ UP = (𝑑 ∈ V, 𝑒 ∈ V ↦ ⦋(Base‘𝑑) / 𝑏⦌⦋(Base‘𝑒) / 𝑐⦌⦋(Hom ‘𝑑) / ℎ⦌⦋(Hom ‘𝑒) / 𝑗⦌⦋(comp‘𝑒) / 𝑜⦌(𝑓 ∈ (𝑑 Func 𝑒), 𝑤 ∈ 𝑐 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑚 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝑏 ∀𝑔 ∈ (𝑤𝑗((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥ℎ𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑜((1st ‘𝑓)‘𝑦))𝑚))})) | ||
| Theorem | reldmup 49430 | The domain of UP is a relation. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ Rel dom UP | ||
| Theorem | upfval 49431* | Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) (Proof shortened by Zhi Wang, 12-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) ⇒ ⊢ (𝐷 UP 𝐸) = (𝑓 ∈ (𝐷 Func 𝐸), 𝑤 ∈ 𝐶 ↦ {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑤𝐽((1st ‘𝑓)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝑓)𝑦)‘𝑘)(⟨𝑤, ((1st ‘𝑓)‘𝑥)⟩𝑂((1st ‘𝑓)‘𝑦))𝑚))}) | ||
| Theorem | upfval2 49432* | Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝑊 ∈ 𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) ⇒ ⊢ (𝜑 → (𝐹(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘𝐹)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘𝐹)𝑦)‘𝑘)(⟨𝑊, ((1st ‘𝐹)‘𝑥)⟩𝑂((1st ‘𝐹)‘𝑦))𝑚))}) | ||
| Theorem | upfval3 49433* | Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝑊 ∈ 𝐶) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) ⇒ ⊢ (𝜑 → (⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚))}) | ||
| Theorem | isuplem 49434* | Lemma for isup 49435 and other theorems. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝑊 ∈ 𝐶) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) ⇒ ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀)))) | ||
| Theorem | isup 49435* | The predicate "is a universal pair". (Contributed by Zhi Wang, 24-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ 𝑂 = (comp‘𝐸) & ⊢ (𝜑 → 𝑊 ∈ 𝐶) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋))) ⇒ ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))) | ||
| Theorem | uppropd 49436 | If two categories have the same set of objects, morphisms, and compositions, then they have the same universal pairs. (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 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 UP 𝐶) = (𝐵 UP 𝐷)) | ||
| Theorem | reldmup2 49437 | The domain of (𝐷 UP 𝐸) is a relation. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ Rel dom (𝐷 UP 𝐸) | ||
| Theorem | relup 49438 | The set of universal pairs is a relation. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ Rel (𝐹(𝐷 UP 𝐸)𝑊) | ||
| Theorem | uprcl 49439 | Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝑋 ∈ (𝐹(𝐷 UP 𝐸)𝑊) → (𝐹 ∈ (𝐷 Func 𝐸) ∧ 𝑊 ∈ 𝐶)) | ||
| Theorem | up1st2nd 49440 | Rewrite the universal property predicate with separated parts. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀) ⇒ ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀) | ||
| Theorem | up1st2ndr 49441 | Combine separated parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀) ⇒ ⊢ (𝜑 → 𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀) | ||
| Theorem | up1st2ndb 49442 | Combine/separate parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) ⇒ ⊢ (𝜑 → (𝑋(𝐹(𝐷 UP 𝐸)𝑊)𝑀 ↔ 𝑋(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐷 UP 𝐸)𝑊)𝑀)) | ||
| Theorem | up1st2nd2 49443 | 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 49444 | 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 49445 | 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 49446 | 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 49447 | 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 49448 | 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 49449* | 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 49450* | 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 49451 | 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 49452 | Lemma for uptpos 49453. (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 49453 | 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 49454 | 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 49455 | 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 49456 | 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 49457 | 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 49458 | 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 49459 | 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 49460 | 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 49461* | Lemma for oppcup3 49464. (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 49462* | 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 49463* | 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 49464* | 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 49465* | Lemma for uptr 49468. (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 49466* | Lemma for uptr 49468. (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 49467 | Lemma for uptr 49468. (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 49468 | 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 49469 | 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 49470 | 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 49471 | 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 49472 | 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 49473 | 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 49474 | 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 49475 | 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 49476 | 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 49477 | 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 49478* | 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 49479 | 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 49480 | 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 49481 | 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 49482 | 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 49483 | 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 49484 | 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 49485 | 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 49486 | 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 49487 | 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 49488 | 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 49489 | 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 49490 | Initial objects are terminal in the opposite category. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ (TermO‘(oppCat‘𝐶))) | ||
| Theorem | oppctermo 49491 | 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 49492 | 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 49493 | 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 49494 | Lemma for initopropdlem 49495, termopropdlem 49496, and zeroopropdlem 49497. (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 49495 | Lemma for initopropd 49498. (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 49496 | Lemma for termopropd 49499. (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 49497 | Lemma for zeroopropd 49500. (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 49498 | 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 49499 | 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 49500 | 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)) | ||
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