HomeTheorem List (p. 502 of 510)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ipolub00 50101 | The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) & ⊢ (𝜑 → ∅ ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑈‘∅) = ∅) | ||
| Theorem | ipoglb0 50102 | The GLB of the empty set is the union of the base. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐹) & ⊢ (𝜑 → 𝐺 = (glb‘𝐼)) & ⊢ (𝜑 → ∪ 𝐹 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐹) | ||
| Theorem | mrelatlubALT 50103 | Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐹 = (mrCls‘𝐶) & ⊢ 𝐿 = (lub‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈)) | ||
| Theorem | mrelatglbALT 50104 | Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈) | ||
| Theorem | mreclat 50105 | A Moore space is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat) | ||
| Theorem | topclat 50106 | A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐼 ∈ CLat) | ||
| Theorem | toplatglb0 50107 | The empty intersection in a topology is realized by the base set. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ (𝜑 → (𝐺‘∅) = ∪ 𝐽) | ||
| Theorem | toplatlub 50108 | Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ 𝐽) & ⊢ 𝑈 = (lub‘𝐼) ⇒ ⊢ (𝜑 → (𝑈‘𝑆) = ∪ 𝑆) | ||
| Theorem | toplatglb 50109 | Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝑆 ⊆ 𝐽) & ⊢ 𝐺 = (glb‘𝐼) & ⊢ (𝜑 → 𝑆 ≠ ∅) ⇒ ⊢ (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆)) | ||
| Theorem | toplatjoin 50110 | Joins in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) & ⊢ ∨ = (join‘𝐼) ⇒ ⊢ (𝜑 → (𝐴 ∨ 𝐵) = (𝐴 ∪ 𝐵)) | ||
| Theorem | toplatmeet 50111 | Meets in a topology are realized by intersections. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) & ⊢ ∧ = (meet‘𝐼) ⇒ ⊢ (𝜑 → (𝐴 ∧ 𝐵) = (𝐴 ∩ 𝐵)) | ||
| Theorem | topdlat 50112 | A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
| ⊢ 𝐼 = (toInc‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat) | ||
| Theorem | elmgpcntrd 50113* | The center of a ring. (Contributed by Zhi Wang, 11-Sep-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (Cntr‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑋(.r‘𝑅)𝑦) = (𝑦(.r‘𝑅)𝑋)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑍) | ||
| Theorem | asclelbasALT 50114 | Alternate proof of asclelbas 22191. (Contributed by Zhi Wang, 11-Sep-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Base‘𝑊)) | ||
| Theorem | asclcntr 50115 | The algebra scalar lifting function maps into the center of the algebra. Equivalently, a lifted scalar is a center 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) & ⊢ 𝑀 = (mulGrp‘𝑊) ⇒ ⊢ (𝜑 → (𝐴‘𝐶) ∈ (Cntr‘𝑀)) | ||
| Theorem | asclcom 50116 |
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 22171, assa2ass2 22172, and asclval 22187, 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 50117 | 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 50118* | Lemma for catprs 50119. (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 50119* | A preorder can be extracted from a category. See catprs2 50120 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 50120* | 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 50121 and catprsc2 50122 for constructions satisfying the hypothesis "catprs.1". See catprs 50119 for a more primitive version. See prsthinc 50572 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 50121* | A construction of the preorder induced by a category. See catprs2 50120 for details. See also catprsc2 50122 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 50122* | An alternate construction of the preorder induced by a category. See catprs2 50120 for details. See also catprsc 50121 for a different construction. The two constructions are different because df-cat 17842 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 50123 | A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 50686 for converting a monoid to a category. Lemma for bj-endmnd 38240. (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 50124 | 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 50125 | Lemma for oppcmndc 50127. 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 50126 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 50126* | 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 50127 | 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 50128 | 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 50129 | 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 50130 | Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑆 = (Sect‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | sectrcl2 50131 | 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 50132 | Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.) |
| ⊢ 𝑁 = (Inv‘𝐶) & ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | invrcl2 50133 | 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 50134 | 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 50135 | 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 50136* | Lemma for upeu2 50280. 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 50137 | 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 50138 | 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 17950 (see isofnALT 50139). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | ||
| Theorem | isofnALT 50139 | 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 50140* | 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 50141 | Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.) |
| ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
| Theorem | isorcl2 50142 | 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 50143 | 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 50144 | Lemma for sectpropd 50145. (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 50145 | 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 50146 | Lemma for invpropd 50147. (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 50147 | 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 50148 | Lemma for isopropd 50149. (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 50149 | 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 50150 | ≃𝑐 is a function on Cat. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ ≃𝑐 Fn Cat | ||
| Theorem | cicrcl2 50151 | Isomorphism implies the structure being a category. (Contributed by Zhi Wang, 26-Oct-2025.) |
| ⊢ (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝐶 ∈ Cat) | ||
| Theorem | oppccic 50152 | 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 50153 | The set of isomorphic objects is a relation. Simplifies cicer 17981 (see cicerALT 50154). (Contributed by Zhi Wang, 27-Oct-2025.) |
| ⊢ (𝐶 ∈ Cat → Rel ( ≃𝑐 ‘𝐶)) | ||
| Theorem | cicerALT 50154 | 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 50155 | 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 50156 | 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 50157 | Lemma for cicpropd 50158. (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 50158 | 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 50159 | 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 50160 | 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 50161 | 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 50162* | Lemma for iinfssc 50165. (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 50163* | Lemma for iinfssc 50165. (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 50164* | Lemma for iinfssc 50165. (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 50165* | 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 50166* | 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 50167* | 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 50168* | 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 50169* | 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 50170* | 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 50171* | Lemma for discsubc 50172. (Contributed by Zhi Wang, 1-Nov-2025.) |
| ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) ⇒ ⊢ 𝐽 Fn (𝑆 × 𝑆) | ||
| Theorem | discsubc 50172* | 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 50173* | Lemma for iinfconstbas 50174. (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 50174* | 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 50175* | Lemma for nelsubc 50176. (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 50176* | 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 50177 | 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 50178* | Lemma for nelsubc3 50179. (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 50179* |
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 17987 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 18024). To construct such a category, see setc1onsubc 50710 and cnelsubc 50712. (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 50180* | 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 50712). (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 50181* | Lemma for resccat 50182. (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 50182* | 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 50183 | The domain of Func is a relation. (Contributed by Zhi Wang, 12-Nov-2025.) |
| ⊢ Rel dom Func | ||
| Theorem | func1st2nd 50184 | 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 50185 | 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 50186 | 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 50187 | Reverse closure for a functor. (Contributed by Zhi Wang, 17-Sep-2025.) |
| ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) ⇒ ⊢ (𝜑 → 𝐷 ∈ Cat) | ||
| Theorem | funcrcl3 50188 | Reverse closure for a functor. (Contributed by Zhi Wang, 17-Sep-2025.) |
| ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) ⇒ ⊢ (𝜑 → 𝐸 ∈ Cat) | ||
| Theorem | funcf2lem 50189* | 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 50190* | 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 50191 | Lemma for 0funcg 50193. (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 50192 | 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 50193 | 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 50194 | Lemma for 0funcALT 50196. (Contributed by Zhi Wang, 7-Oct-2025.) |
| ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏))) & ⊢ (𝜒 ↔ 𝜂) & ⊢ (𝜃 ↔ 𝜁) & ⊢ 𝜏 ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜂 ∧ 𝜁))) | ||
| Theorem | 0func 50195 | 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 50196 | Alternate proof of 0func 50195. (Contributed by Zhi Wang, 7-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → (∅ Func 𝐶) = {⟨∅, ∅⟩}) | ||
| Theorem | func0g 50197 | 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 50198 | 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 50199* | 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 50200 | 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)⟩)) | ||
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