HomeTheorem List (p. 491 of 498)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | asclcom 49001 |
Scalars are commutative after being lifted.
However, the scalars themselves are not necessarily commutative if the algebra is not a faithful module. For example, Let 𝐹 be the 2 by 2 upper triangular matrix algebra over a commutative ring 𝑊. It is provable that 𝐹 is in general non-commutative. Define scalar multiplication 𝐶 · 𝑋 as multipying the top-left entry, which is a "vector" element of 𝑊, of the "scalar" 𝐶, which is now an upper triangular matrix, with the "vector" 𝑋 ∈ (Base‘𝑊). Equivalently, the algebra scalar lifting function is not necessarily injective unless the algebra is faithful. Therefore, all "scalar injection" was renamed. Alternate proof involves assa2ass 21779, assa2ass2 21780, and asclval 21796, by setting 𝑋 and 𝑌 the multiplicative identity of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) |
| ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ∗ = (.r‘𝐹) & ⊢ (𝜑 → 𝐷 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶))) | ||
| Theorem | homf0 49002 | The base is empty iff the functionalized Hom-set operation is empty. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ ((Base‘𝐶) = ∅ ↔ (Homf ‘𝐶) = ∅) | ||
| Theorem | catprslem 49003* | Lemma for catprs 49004. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) | ||
| Theorem | catprs 49004* | A preorder can be extracted from a category. See catprs2 49005 for more details. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))) | ||
| Theorem | catprs2 49005* | A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 49006 and catprsc2 49007 for constructions satisfying the hypothesis "catprs.1". See catprs 49004 for a more primitive version. See prsthinc 49457 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → ≤ = (le‘𝐶)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Proset ) | ||
| Theorem | catprsc 49006* | A construction of the preorder induced by a category. See catprs2 49005 for details. See also catprsc2 49007 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.) |
| ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)}) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)) | ||
| Theorem | catprsc2 49007* | An alternate construction of the preorder induced by a category. See catprs2 49005 for details. See also catprsc 49006 for a different construction. The two constructions are different because df-cat 17636 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.) |
| ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅}) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅)) | ||
| Theorem | endmndlem 49008 | A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 49571 for converting a monoid to a category. Lemma for bj-endmnd 37313. (Contributed by Zhi Wang, 25-Sep-2024.) |
| ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ Cat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀)) & ⊢ (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘𝑀)) ⇒ ⊢ (𝜑 → 𝑀 ∈ Mnd) | ||
| Theorem | oppccatb 49009 | An opposite category is a category. (Contributed by Zhi Wang, 23-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝑂 ∈ Cat)) | ||
| Theorem | oppcmndclem 49010 | Lemma for oppcmndc 49012. Everything is true for two distinct elements in a singleton or an empty set (since it is impossible). Note that if this theorem and oppcendc 49011 are in ¬ 𝑥 = 𝑦 form, then both proofs should be one step shorter. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ (𝜑 → 𝐵 = {𝐴}) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≠ 𝑌 → 𝜓)) | ||
| Theorem | oppcendc 49011* | The opposite category of a category whose morphisms are all endomorphisms has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025.) |
| ⊢ 𝑂 = (oppCat‘𝐶) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≠ 𝑦 → (𝑥𝐻𝑦) = ∅)) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝑂)) | ||
| Theorem | oppcmndc 49012 | 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 49013 | 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 49014 | 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 49015 | Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | sectrcl2 49016 | 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 49017 | Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | invrcl2 49018 | 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 49019 | 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 49020 | 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 49021* | Lemma for upeu2 49165. 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 49022 | 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 49023 | 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 17744 (see isofnALT 49024). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | isofnALT 49024 | 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 49025* | 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 49026 | Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | isorcl2 49027 | 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 49028 | 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 49029 | Lemma for sectpropd 49030. (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 49030 | 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 49031 | Lemma for invpropd 49032. (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 49032 | 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 49033 | Lemma for isopropd 49034. (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 49034 | 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 49035 | ≃𝑐 is a function on Cat. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ ≃𝑐 Fn Cat | ||
| Theorem | cicrcl2 49036 | Isomorphism implies the structure being a category. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝐶 ∈ Cat) | ||
| Theorem | oppccic 49037 | 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 49038 | The set of isomorphic objects is a relation. Simplifies cicer 17775 (see cicerALT 49039). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶)) | ||
| Theorem | cicerALT 49039 | 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 49040 | 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 49041 | 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 49042 | Lemma for cicpropd 49043. (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 49043 | 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 49044 | 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 49045 | 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 49046 | 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 49047* | Lemma for iinfssc 49050. (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 49048* | Lemma for iinfssc 49050. (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 49049* | Lemma for iinfssc 49050. (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 49050* | 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 49051* | 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 49052* | 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 49053* | 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 49054* | 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 49055* | 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 49056* | Lemma for discsubc 49057. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) ⇒ ⊢ 𝐽 Fn (𝑆 × 𝑆) | ||
| Theorem | discsubc 49057* | 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 49058* | Lemma for iinfconstbas 49059. (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 49059* | 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 49060* | Lemma for nelsubc 49061. (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 49061* | 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 49062 | 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 49063* | Lemma for nelsubc3 49064. (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 49064* |
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 17781 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 17818). To construct such a category, see setc1onsubc 49595 and cnelsubc 49597. (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 49065* | 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 49597). (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 49066* | Lemma for resccat 49067. (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 49067* | 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 49068 | The domain of Func is a relation. (Contributed by Zhi Wang, 12-Nov-2025.) |
| ⊢ Rel dom Func | ||
| Theorem | func1st2nd 49069 | 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 49070 | 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 49071 | 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 49072 | Reverse closure for a functor. (Contributed by Zhi Wang, 17-Sep-2025.) |
| ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) ⇒ ⊢ (𝜑 → 𝐷 ∈ Cat) | ||
| Theorem | funcrcl3 49073 | Reverse closure for a functor. (Contributed by Zhi Wang, 17-Sep-2025.) |
| ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) ⇒ ⊢ (𝜑 → 𝐸 ∈ Cat) | ||
| Theorem | funcf2lem 49074* | 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 49075* | 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 49076 | Lemma for 0funcg 49078. (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 49077 | 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 49078 | 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 49079 | Lemma for 0funcALT 49081. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) & ⊢ (𝜒 ↔ 𝜂) & ⊢ (𝜃 ↔ 𝜁) & ⊢ 𝜏 ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜂 ∧ 𝜁))) | ||
| Theorem | 0func 49080 | 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 49081 | Alternate proof of 0func 49080. (Contributed by Zhi Wang, 7-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩}) | ||
| Theorem | func0g 49082 | 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 49083 | 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 49084* | 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 49085 | 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 49086 | 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 49087 | 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 49088 | 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 49089 | 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 49090 | 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 49091 | Reverse closure for an identity functor. (Contributed by Zhi Wang, 10-Nov-2025.) |
| ⊢ ((idfunc‘𝐶) ∈ (𝐷 Func 𝐸) → 𝐶 ∈ Cat) | ||
| Theorem | idfu1stf1o 49092 | 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 49093 | Lemma for idfu1sta 49094. (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 49094 | 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 49095 | 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 49096 | 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 49097* | Lemma for imasubc 49144. (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 49098* | Lemma for imasubc 49144. (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 49099* | Lemma for imasubc 49144. (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 49100 | Lemma for imaf1hom 49101 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)) | ||
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