HomeTheorem List (p. 491 of 498)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | oppcmndc 49001 | The opposite category of a category whose base set is a singleton or an empty set has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐵 = {𝑋}) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂)) | ||
| Theorem | idmon 49002 | An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋)) | ||
| Theorem | idepi 49003 | An identity arrow, or an identity morphism, is an epimorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐸 = (Epi‘𝐶) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐸𝑋)) | ||
| Theorem | sectrcl 49004 | Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | sectrcl2 49005 | Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | invrcl 49006 | Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | invrcl2 49007 | Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | isinv2 49008 | The property "𝐹 is an inverse of 𝐺". (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹)) | ||
| Theorem | isisod 49009 | The predicate "is an isomorphism" (deduction form). (Contributed by Zhi Wang, 16-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋)) & ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)) & ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | ||
| Theorem | upeu2lem 49010* | Lemma for upeu2 49154. There exists a unique morphism from 𝑌 to 𝑍 that commutes if 𝐹:𝑋⟶𝑌 is an isomorphism. (Contributed by Zhi Wang, 20-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑍)) ⇒ ⊢ (𝜑 → ∃!𝑘 ∈ (𝑌𝐻𝑍)𝐺 = (𝑘(⟨𝑋, 𝑌⟩ · 𝑍)𝐹)) | ||
| Theorem | sectfn 49011 | The function value of the function returning the sections of a category is a function over the Cartesian square of the base set of the category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → (Sect‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | invfn 49012 | The function value of the function returning the inverses of a category is a function over the Cartesian square of the base set of the category. Simplifies isofn 17717 (see isofnALT 49013). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | isofnALT 49013 | The function value of the function returning the isomorphisms of a category is a function over the Cartesian square of the base set of the category. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | isofval2 49014* | Function value of the function returning the isomorphisms of a category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦))) | ||
| Theorem | isorcl 49015 | Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | isorcl2 49016 | Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) | ||
| Theorem | isoval2 49017 | The isomorphisms are the domain of the inverse relation. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌) | ||
| Theorem | sectpropdlem 49018 | Lemma for sectpropd 49019. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Sect‘𝐶)) → 𝑃 ∈ (Sect‘𝐷)) | ||
| Theorem | sectpropd 49019 | Two structures with the same base, hom-sets and composition operation have the same sections. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Sect‘𝐶) = (Sect‘𝐷)) | ||
| Theorem | invpropdlem 49020 | Lemma for invpropd 49021. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Inv‘𝐶)) → 𝑃 ∈ (Inv‘𝐷)) | ||
| Theorem | invpropd 49021 | Two structures with the same base, hom-sets and composition operation have the same inverses. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Inv‘𝐶) = (Inv‘𝐷)) | ||
| Theorem | isopropdlem 49022 | Lemma for isopropd 49023. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐷)) | ||
| Theorem | isopropd 49023 | Two structures with the same base, hom-sets and composition operation have the same isomorphisms. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → (Iso‘𝐶) = (Iso‘𝐷)) | ||
| Theorem | cicfn 49024 | ≃𝑐 is a function on Cat. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ ≃𝑐 Fn Cat | ||
| Theorem | cicrcl2 49025 | Isomorphism implies the structure being a category. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝐶 ∈ Cat) | ||
| Theorem | oppccic 49026 | Isomorphic objects are isomorphic in the opposite category. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝑅( ≃𝑐 ‘𝐶)𝑆) ⇒ ⊢ (𝜑 → 𝑅( ≃𝑐 ‘𝑂)𝑆) | ||
| Theorem | relcic 49027 | The set of isomorphic objects is a relation. Simplifies cicer 17748 (see cicerALT 49028). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶)) | ||
| Theorem | cicerALT 49028 | Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.) (Proof shortened by Zhi Wang, 3-Nov-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) Er (Base‘𝐶)) | ||
| Theorem | cic1st2nd 49029 | Reconstruction of a pair of isomorphic objects in terms of its ordered pair components. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩) | ||
| Theorem | cic1st2ndbr 49030 | Rewrite the predicate of isomorphic objects with separated parts. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝑃 ∈ ( ≃𝑐 ‘𝐶) → (1st ‘𝑃)( ≃𝑐 ‘𝐶)(2nd ‘𝑃)) | ||
| Theorem | cicpropdlem 49031 | Lemma for cicpropd 49032. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ ((𝜑 ∧ 𝑃 ∈ ( ≃𝑐 ‘𝐶)) → 𝑃 ∈ ( ≃𝑐 ‘𝐷)) | ||
| Theorem | cicpropd 49032 | Two structures with the same base, hom-sets and composition operation have the same isomorphic objects. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷)) & ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷)) ⇒ ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝐷)) | ||
| Theorem | oppccicb 49033 | Isomorphic objects are isomorphic in the opposite category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅( ≃𝑐 ‘𝑂)𝑆) | ||
| Theorem | oppcciceq 49034 | The opposite category has the same isomorphic objects as the original category. (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝑂) | ||
| Theorem | dmdm 49035 | The double domain of a function on a Cartesian square. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ (𝐴 Fn (𝐵 × 𝐵) → 𝐵 = dom dom 𝐴) | ||
| Theorem | iinfssclem1 49036* | Lemma for iinfssc 49039. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻) & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → 𝐾 = (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆, 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 (𝑧𝐻𝑤))) | ||
| Theorem | iinfssclem2 49037* | Lemma for iinfssc 49039. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻) & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → 𝐾 Fn (∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆)) | ||
| Theorem | iinfssclem3 49038* | Lemma for iinfssc 49039. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = dom dom 𝐻) & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑥 ∈ 𝐴 𝑆) & ⊢ (𝜑 → 𝑌 ∈ ∩ 𝑥 ∈ 𝐴 𝑆) ⇒ ⊢ (𝜑 → (𝑋𝐾𝑌) = ∩ 𝑥 ∈ 𝐴 (𝑋𝐻𝑌)) | ||
| Theorem | iinfssc 49039* | Indexed intersection of subcategories is a subcategory (the category-agnostic version). (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ⊆cat 𝐽) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) ⇒ ⊢ (𝜑 → 𝐾 ⊆cat 𝐽) | ||
| Theorem | iinfsubc 49040* | Indexed intersection of subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻 ∈ (Subcat‘𝐶)) & ⊢ (𝜑 → 𝐾 = (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 (𝐻‘𝑦))) ⇒ ⊢ (𝜑 → 𝐾 ∈ (Subcat‘𝐶)) | ||
| Theorem | iinfprg 49041* | Indexed intersection of functions with an unordered pair index. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) = (𝑥 ∈ ∩ 𝑦 ∈ {𝐴, 𝐵}dom 𝑦 ↦ ∩ 𝑦 ∈ {𝐴, 𝐵} (𝑦‘𝑥))) | ||
| Theorem | infsubc 49042* | The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom 𝐴 ∩ dom 𝐵) ↦ ((𝐴‘𝑥) ∩ (𝐵‘𝑥))) ∈ (Subcat‘𝐶)) | ||
| Theorem | infsubc2 49043* | The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ ((𝐴 ∈ (Subcat‘𝐶) ∧ 𝐵 ∈ (Subcat‘𝐶)) → (𝑥 ∈ (dom dom 𝐴 ∩ dom dom 𝐵), 𝑦 ∈ (dom dom 𝐴 ∩ dom dom 𝐵) ↦ ((𝑥𝐴𝑦) ∩ (𝑥𝐵𝑦))) ∈ (Subcat‘𝐶)) | ||
| Theorem | infsubc2d 49044* | The intersection of two subcategories is a subcategory. (Contributed by Zhi Wang, 31-Oct-2025.) |
| ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) & ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) & ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) & ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝑆 ∩ 𝑇), 𝑦 ∈ (𝑆 ∩ 𝑇) ↦ ((𝑥𝐻𝑦) ∩ (𝑥𝐽𝑦))) ∈ (Subcat‘𝐶)) | ||
| Theorem | discsubclem 49045* | Lemma for discsubc 49046. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) ⇒ ⊢ 𝐽 Fn (𝑆 × 𝑆) | ||
| Theorem | discsubc 49046* | A discrete category, whose only morphisms are the identity morphisms, is a subcategory. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐼 = (Id‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | ||
| Theorem | iinfconstbaslem 49047* | Lemma for iinfconstbas 49048. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐼 = (Id‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)})) ⇒ ⊢ (𝜑 → 𝐽 ∈ 𝐴) | ||
| Theorem | iinfconstbas 49048* | The discrete category is the indexed intersection of all subcategories with the same base. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐼 = (Id‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝐴 = ((Subcat‘𝐶) ∩ {𝑗 ∣ 𝑗 Fn (𝑆 × 𝑆)})) ⇒ ⊢ (𝜑 → 𝐽 = (𝑧 ∈ ∩ ℎ ∈ 𝐴 dom ℎ ↦ ∩ ℎ ∈ 𝐴 (ℎ‘𝑧))) | ||
| Theorem | nelsubclem 49049* | Lemma for nelsubc 49050. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑆 ≠ ∅) & ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅})) & ⊢ 𝐻 = (Homf ‘𝐶) ⇒ ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 𝐼 ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)𝜓)))) | ||
| Theorem | nelsubc 49050* | An empty "hom-set" for non-empty base satisfies all conditions for a subcategory but the existence of identity morphisms. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑆 ≠ ∅) & ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅})) & ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat 𝐻 ∧ (¬ ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))) | ||
| Theorem | nelsubc2 49051 | An empty "hom-set" for non-empty base is not a subcategory. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑆 ≠ ∅) & ⊢ (𝜑 → 𝐽 = ((𝑆 × 𝑆) × {∅})) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → ¬ 𝐽 ∈ (Subcat‘𝐶)) | ||
| Theorem | nelsubc3lem 49052* | Lemma for nelsubc3 49053. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ 𝐶 ∈ Cat & ⊢ 𝐽 ∈ V & ⊢ 𝑆 ∈ V & ⊢ (𝐽 Fn (𝑆 × 𝑆) ∧ (𝐽 ⊆cat (Homf ‘𝐶) ∧ (¬ ∀𝑥 ∈ 𝑆 ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))) ⇒ ⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ (¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))) | ||
| Theorem | nelsubc3 49053* |
Remark 4.2(2) of [Adamek] p. 48. There exists
a set satisfying all
conditions for a subcategory but the existence of identity morphisms.
Therefore such condition in df-subc 17754 is necessary.
Note that this theorem cheated a little bit because (𝐶 ↾cat 𝐽) is not a category. In fact (𝐶 ↾cat 𝐽) ∈ Cat is a stronger statement than the condition (d) of Definition 4.1(1) of [Adamek] p. 48, as stated here (see the proof of issubc3 17791). To construct such a category, see setc1onsubc 49584 and cnelsubc 49586. (Contributed by Zhi Wang, 5-Nov-2025.) |
| ⊢ ∃𝑐 ∈ Cat ∃𝑗∃𝑠(𝑗 Fn (𝑠 × 𝑠) ∧ (𝑗 ⊆cat (Homf ‘𝑐) ∧ (¬ ∀𝑥 ∈ 𝑠 ((Id‘𝑐)‘𝑥) ∈ (𝑥𝑗𝑥) ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥𝑗𝑦)∀𝑔 ∈ (𝑦𝑗𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑗𝑧)))) | ||
| Theorem | ssccatid 49054* | A category 𝐶 restricted by 𝐽 is a category if all of the following are satisfied: a) the base is a subset of base of 𝐶, b) all hom-sets are subsets of hom-sets of 𝐶, c) it has identity morphisms for all objects, d) the composition under 𝐶 is closed in 𝐽. But 𝐽 might not be a subcategory of 𝐶 (see cnelsubc 49586). (Contributed by Zhi Wang, 6-Nov-2025.) |
| ⊢ 𝐻 = (Homf ‘𝐶) & ⊢ 𝐷 = (𝐶 ↾cat 𝐽) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐽 ⊆cat 𝐻) & ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 1 ∈ (𝑦𝐽𝑦)) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → ( 1 (⟨𝑎, 𝑏⟩ · 𝑏)𝑚) = 𝑚) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑚 ∈ (𝑎𝐽𝑏))) → (𝑚(⟨𝑎, 𝑎⟩ · 𝑏) 1 ) = 𝑚) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)) ⇒ ⊢ (𝜑 → (𝐷 ∈ Cat ∧ (Id‘𝐷) = (𝑦 ∈ 𝑆 ↦ 1 ))) | ||
| Theorem | resccatlem 49055* | Lemma for resccat 49056. (Contributed by Zhi Wang, 6-Nov-2025.) |
| ⊢ 𝐷 = (𝐶 ↾cat 𝐽) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑆 = (Base‘𝐸) & ⊢ 𝐽 = (Homf ‘𝐸) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐸) & ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat)) | ||
| Theorem | resccat 49056* | A class 𝐶 restricted by the hom-sets of another set 𝐸, whose base is a subset of the base of 𝐶 and whose composition is compatible with 𝐶, is a category iff 𝐸 is a category. Note that the compatibility condition "resccat.1" can be weakened by removing 𝑥 ∈ 𝑆 because 𝑓 ∈ (𝑥𝐽𝑦) implies these. (Contributed by Zhi Wang, 6-Nov-2025.) |
| ⊢ 𝐷 = (𝐶 ↾cat 𝐽) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑆 = (Base‘𝐸) & ⊢ 𝐽 = (Homf ‘𝐸) & ⊢ · = (comp‘𝐶) & ⊢ ∙ = (comp‘𝐸) & ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ (𝑓 ∈ (𝑥𝐽𝑦) ∧ 𝑔 ∈ (𝑦𝐽𝑧))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ ∙ 𝑧)𝑓)) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐷 ∈ Cat ↔ 𝐸 ∈ Cat)) | ||
| Theorem | reldmfunc 49057 | The domain of Func is a relation. (Contributed by Zhi Wang, 12-Nov-2025.) |
| ⊢ Rel dom Func | ||
| Theorem | func1st2nd 49058 | Rewrite the functor predicate with separated parts. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) ⇒ ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) | ||
| Theorem | func1st 49059 | Extract the first member of a functor. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) ⇒ ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹) | ||
| Theorem | func2nd 49060 | Extract the second member of a functor. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) ⇒ ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺) | ||
| Theorem | funcrcl2 49061 | Reverse closure for a functor. (Contributed by Zhi Wang, 17-Sep-2025.) |
| ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) ⇒ ⊢ (𝜑 → 𝐷 ∈ Cat) | ||
| Theorem | funcrcl3 49062 | Reverse closure for a functor. (Contributed by Zhi Wang, 17-Sep-2025.) |
| ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) ⇒ ⊢ (𝜑 → 𝐸 ∈ Cat) | ||
| Theorem | funcf2lem 49063* | A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 1-Oct-2024.) |
| ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) | ||
| Theorem | funcf2lem2 49064* | A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ 𝐵 = (𝐸‘𝐶) ⇒ ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) | ||
| Theorem | 0funcglem 49065 | Lemma for 0funcg 49067. (Contributed by Zhi Wang, 17-Oct-2025.) |
| ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) & ⊢ (𝜑 → (𝜒 ↔ 𝜂)) & ⊢ (𝜑 → (𝜃 ↔ 𝜁)) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜂 ∧ 𝜁))) | ||
| Theorem | 0funcg2 49066 | The functor from the empty category. (Contributed by Zhi Wang, 17-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ∅ = (Base‘𝐶)) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (𝐹(𝐶 Func 𝐷)𝐺 ↔ (𝐹 = ∅ ∧ 𝐺 = ∅))) | ||
| Theorem | 0funcg 49067 | The functor from the empty category. Corollary of Definition 3.47 of [Adamek] p. 40, Definition 7.1 of [Adamek] p. 101, Example 3.3(4.c) of [Adamek] p. 24, and Example 7.2(3) of [Adamek] p. 101. (Contributed by Zhi Wang, 17-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ∅ = (Base‘𝐶)) & ⊢ (𝜑 → 𝐷 ∈ Cat) ⇒ ⊢ (𝜑 → (𝐶 Func 𝐷) = {⟨∅, ∅⟩}) | ||
| Theorem | 0funclem 49068 | Lemma for 0funcALT 49070. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) & ⊢ (𝜒 ↔ 𝜂) & ⊢ (𝜃 ↔ 𝜁) & ⊢ 𝜏 ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜂 ∧ 𝜁))) | ||
| Theorem | 0func 49069 | The functor from the empty category. (Contributed by Zhi Wang, 7-Oct-2025.) (Proof shortened by Zhi Wang, 17-Oct-2025.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩}) | ||
| Theorem | 0funcALT 49070 | Alternate proof of 0func 49069. (Contributed by Zhi Wang, 7-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩}) | ||
| Theorem | func0g 49071 | The source category of a functor to the empty category must be empty as well. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐵 = ∅) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) ⇒ ⊢ (𝜑 → 𝐴 = ∅) | ||
| Theorem | func0g2 49072 | The source category of a functor to the empty category must be empty as well. (Contributed by Zhi Wang, 19-Oct-2025.) |
| ⊢ 𝐴 = (Base‘𝐶) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐵 = ∅) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) ⇒ ⊢ (𝜑 → 𝐴 = ∅) | ||
| Theorem | initc 49073* | Sets with empty base are the only initial objects in the category of small categories. Example 7.2(3) of [Adamek] p. 101. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ ((𝐶 ∈ V ∧ ∅ = (Base‘𝐶)) ↔ ∀𝑑 ∈ Cat ∃!𝑓 𝑓 ∈ (𝐶 Func 𝑑)) | ||
| Theorem | cofu1st2nd 49074 | Rewrite the functor composition with separated functor parts. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) & ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸)) ⇒ ⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)) | ||
| Theorem | rescofuf 49075 | The restriction of functor composition is a function from product functor space to functor space. (Contributed by Zhi Wang, 25-Sep-2025.) |
| ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸) | ||
| Theorem | cofu1a 49076 | Value of the object part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) & ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿) & ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = (𝑀‘𝑋)) | ||
| Theorem | cofu2a 49077 | Value of the morphism part of the functor composition. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) & ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿) & ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = ⟨𝑀, 𝑁⟩) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑋𝑁𝑌)‘𝑅)) | ||
| Theorem | cofucla 49078 | The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) & ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿) ⇒ ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) ∈ (𝐶 Func 𝐸)) | ||
| Theorem | funchomf 49079 | Source categories of a functor have the same set of objects and morphisms. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺) & ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺) ⇒ ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵)) | ||
| Theorem | idfurcl 49080 | Reverse closure for an identity functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ ((idfunc‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat) | ||
| Theorem | idfu1stf1o 49081 | The identity functor/inclusion functor is bijective on objects. (Contributed by Zhi Wang, 16-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝐶 ∈ Cat → (1st ‘𝐼):𝐵–1-1-onto→𝐵) | ||
| Theorem | idfu1stalem 49082 | Lemma for idfu1sta 49083. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | ||
| Theorem | idfu1sta 49083 | Value of the object part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) ⇒ ⊢ (𝜑 → (1st ‘𝐼) = ( I ↾ 𝐵)) | ||
| Theorem | idfu1a 49084 | Value of the object part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((1st ‘𝐼)‘𝑋) = 𝑋) | ||
| Theorem | idfu2nda 49085 | Value of the morphism part of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐷)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (𝑋(Hom ‘𝐷)𝑌)) ⇒ ⊢ (𝜑 → (𝑋(2nd ‘𝐼)𝑌) = ( I ↾ 𝐻)) | ||
| Theorem | imasubclem1 49086* | Lemma for imasubc 49133. (Contributed by Zhi Wang, 6-Nov-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷) ∈ V) | ||
| Theorem | imasubclem2 49087* | Lemma for imasubc 49133. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ 𝐾 = (𝑦 ∈ 𝑋, 𝑧 ∈ 𝑌 ↦ ∪ 𝑥 ∈ ((◡𝐹 “ 𝐴) × (◡𝐺 “ 𝐵))((𝐻‘𝐶) “ 𝐷)) ⇒ ⊢ (𝜑 → 𝐾 Fn (𝑋 × 𝑌)) | ||
| Theorem | imasubclem3 49088* | Lemma for imasubc 49133. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐾 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ∪ 𝑧 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐺 “ {𝑦}))((𝐻‘𝐶) “ 𝐷)) ⇒ ⊢ (𝜑 → (𝑋𝐾𝑌) = ∪ 𝑧 ∈ ((◡𝐹 “ {𝑋}) × (◡𝐺 “ {𝑌}))((𝐻‘𝐶) “ 𝐷)) | ||
| Theorem | imaf1homlem 49089 | Lemma for imaf1hom 49090 and other theorems. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ 𝑆 = (𝐹 “ 𝐴) & ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) ⇒ ⊢ (𝜑 → ({(◡𝐹‘𝑋)} = (◡𝐹 “ {𝑋}) ∧ (𝐹‘(◡𝐹‘𝑋)) = 𝑋 ∧ (◡𝐹‘𝑋) ∈ 𝐵)) | ||
| Theorem | imaf1hom 49090* | The hom-set of an image of a functor injective on objects. (Contributed by Zhi Wang, 7-Nov-2025.) |
| ⊢ 𝑆 = (𝐹 “ 𝐴) & ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡𝐹 “ {𝑥}) × (◡𝐹 “ {𝑦}))((𝐺‘𝑝) “ (𝐻‘𝑝))) ⇒ ⊢ (𝜑 → (𝑋𝐾𝑌) = (((◡𝐹‘𝑋)𝐺(◡𝐹‘𝑌)) “ ((◡𝐹‘𝑋)𝐻(◡𝐹‘𝑌)))) | ||
| Theorem | imaidfu2lem 49091 | Lemma for imaidfu2 49093. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸)) ⇒ ⊢ (𝜑 → ((1st ‘𝐼) “ (Base‘𝐷)) = (Base‘𝐷)) | ||
| Theorem | imaidfu 49092* | The image of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸)) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Homf ‘𝐷) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝))) & ⊢ 𝑆 = ((1st ‘𝐼) “ 𝐴) ⇒ ⊢ (𝜑 → (𝐽 ↾ (𝑆 × 𝑆)) = 𝐾) | ||
| Theorem | imaidfu2 49093* | The image of the identity functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 Func 𝐸)) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Homf ‘𝐷) & ⊢ 𝐾 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ∪ 𝑝 ∈ ((◡(1st ‘𝐼) “ {𝑥}) × (◡(1st ‘𝐼) “ {𝑦}))(((2nd ‘𝐼)‘𝑝) “ (𝐻‘𝑝))) & ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) ⇒ ⊢ (𝜑 → 𝐽 = 𝐾) | ||
| Theorem | cofid1a 49094 | Express the object part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷)) & ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼) ⇒ ⊢ (𝜑 → ((1st ‘𝐺)‘((1st ‘𝐹)‘𝑋)) = 𝑋) | ||
| Theorem | cofid2a 49095 | Express the morphism part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷)) & ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌))‘((𝑋(2nd ‘𝐹)𝑌)‘𝑅)) = 𝑅) | ||
| Theorem | cofid1 49096 | Express the object part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿) & ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼) ⇒ ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) = 𝑋) | ||
| Theorem | cofid2 49097 | Express the morphism part of (𝐺 ∘func 𝐹) = 𝐼 explicitly. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) & ⊢ (𝜑 → 𝐾(𝐸 Func 𝐷)𝐿) & ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘func ⟨𝐹, 𝐺⟩) = 𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = 𝑅) | ||
| Theorem | cofidvala 49098* | The property "𝐹 is a section of 𝐺 " in a category of small categories (in a universe); expressed explicitly. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷)) & ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼) & ⊢ 𝐻 = (Hom ‘𝐷) ⇒ ⊢ (𝜑 → (((1st ‘𝐺) ∘ (1st ‘𝐹)) = ( I ↾ 𝐵) ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) = (𝑧 ∈ (𝐵 × 𝐵) ↦ ( I ↾ (𝐻‘𝑧))))) | ||
| Theorem | cofidf2a 49099 | If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the morphism part of 𝐹 is injective, and the morphism part of 𝐺 is surjective in the image of 𝐹. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷)) & ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋(2nd ‘𝐹)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘𝐹)‘𝑋)𝐽((1st ‘𝐹)‘𝑌)) ∧ (((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)):(((1st ‘𝐹)‘𝑋)𝐽((1st ‘𝐹)‘𝑌))–onto→(𝑋𝐻𝑌))) | ||
| Theorem | cofidf1a 49100 | If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the object part of 𝐹 is injective, and the object part of 𝐺 is surjective. (Contributed by Zhi Wang, 15-Nov-2025.) |
| ⊢ 𝐼 = (idfunc‘𝐷) & ⊢ 𝐵 = (Base‘𝐷) & ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸)) & ⊢ (𝜑 → 𝐺 ∈ (𝐸 Func 𝐷)) & ⊢ (𝜑 → (𝐺 ∘func 𝐹) = 𝐼) & ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝜑 → ((1st ‘𝐹):𝐵–1-1→𝐶 ∧ (1st ‘𝐺):𝐶–onto→𝐵)) | ||
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