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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | toplatglb0 45901 | 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 45902 | 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 45903 | 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 45904 | 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 45905 | 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 45906 | A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.) |
⊢ 𝐼 = (toInc‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐼 ∈ DLat) | ||
Theorem | catprslem 45907* | Lemma for catprs 45908. (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 45908* | A preorder can be extracted from a category. See catprs2 45909 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 45909* | 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 45910 and catprsc2 45911 for constructions satisfying the hypothesis "catprs.1". See catprs 45908 for a more primitive version. See prsthinc 45951 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 45910* | A construction of the preorder induced by a category. See catprs2 45909 for details. See also catprsc2 45911 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 45911* | An alternate construction of the preorder induced by a category. See catprs2 45909 for details. See also catprsc 45910 for a different construction. The two constructions are different because df-cat 17125 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 45912 | A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 45979 for converting a monoid to a category. Lemma for bj-endmnd 35172. (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 | idmon 45913 | 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 45914 | 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 | funcf2lem 45915* | 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)))) | ||
Syntax | cthinc 45916 | Extend class notation with the class of thin categories. |
class ThinCat | ||
Definition | df-thinc 45917* | Definition of the class of thin categories, or posetal categories, whose hom-sets each contain at most one morphism. Example 3.26(2) of [Adamek] p. 33. "ThinCat" was taken instead of "PosCat" because the latter might mean the category of posets. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)} | ||
Theorem | isthinc 45918* | The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))) | ||
Theorem | isthinc2 45919* | A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1o. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o)) | ||
Theorem | isthinc3 45920* | A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔)) | ||
Theorem | thincc 45921 | A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat) | ||
Theorem | thinccd 45922 | A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
Theorem | thincssc 45923 | A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ThinCat ⊆ Cat | ||
Theorem | isthincd2lem1 45924* | Lemma for isthincd2 45935 and thincmo2 45925. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | thincmo2 45925 | Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌)) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐶 ∈ ThinCat) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | thincmo 45926* | There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | ||
Theorem | thincmoALT 45927* | Alternate proof for thincmo 45926. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | ||
Theorem | thincmod 45928* | At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | ||
Theorem | thincn0eu 45929* | In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | ||
Theorem | thincid 45930 | In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋)) ⇒ ⊢ (𝜑 → 𝐹 = ( 1 ‘𝑋)) | ||
Theorem | thincmon 45931 | In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 45991. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑀 = (Mono‘𝐶) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌)) | ||
Theorem | thincepi 45932 | In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 45992. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐸 = (Epi‘𝐶) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌)) | ||
Theorem | isthincd2lem2 45933* | Lemma for isthincd2 45935. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐻𝑍)) | ||
Theorem | isthincd 45934* | The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) & ⊢ (𝜑 → 𝐶 ∈ Cat) ⇒ ⊢ (𝜑 → 𝐶 ∈ ThinCat) | ||
Theorem | isthincd2 45935* | The predicate "𝐶 is a thin category" without knowing 𝐶 is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)))) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦)) & ⊢ ((𝜑 ∧ 𝜓) → (𝑔(〈𝑥, 𝑦〉 · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 ))) | ||
Theorem | oppcthin 45936 | The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.) |
⊢ 𝑂 = (oppCat‘𝐶) ⇒ ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat) | ||
Theorem | subthinc 45937 | A subcategory of a thin category is thin. (Contributed by Zhi Wang, 30-Sep-2024.) |
⊢ 𝐷 = (𝐶 ↾cat 𝐽) & ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ ThinCat) ⇒ ⊢ (𝜑 → 𝐷 ∈ ThinCat) | ||
Theorem | functhinclem1 45938* | Lemma for functhinc 45942. Given the object part, there is only one possible morphism part such that the mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.) |
⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ ThinCat) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) & ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅)) ⇒ ⊢ (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ↔ 𝐺 = 𝐾)) | ||
Theorem | functhinclem2 45939* | Lemma for functhinc 45942. (Contributed by Zhi Wang, 1-Oct-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅)) ⇒ ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅)) | ||
Theorem | functhinclem3 45940* | Lemma for functhinc 45942. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.) |
⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))) & ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅)) & ⊢ (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) ⇒ ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) | ||
Theorem | functhinclem4 45941* | Lemma for functhinc 45942. Other requirements on the morphism part are automatically satisfied. (Contributed by Zhi Wang, 1-Oct-2024.) |
⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐸 ∈ ThinCat) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅)) & ⊢ 1 = (Id‘𝐷) & ⊢ 𝐼 = (Id‘𝐸) & ⊢ · = (comp‘𝐷) & ⊢ 𝑂 = (comp‘𝐸) ⇒ ⊢ ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘( 1 ‘𝑎)) = (𝐼‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑚 ∈ (𝑎𝐻𝑏)∀𝑛 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑛(〈𝑎, 𝑏〉 · 𝑐)𝑚)) = (((𝑏𝐺𝑐)‘𝑛)(〈(𝐹‘𝑎), (𝐹‘𝑏)〉𝑂(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑚)))) | ||
Theorem | functhinc 45942* | A functor to a thin category is determined entirely by the object part. The hypothesis "functhinc.1" is related to a monotone function if preorders induced by the categories are considered (catprs2 45909). (Contributed by Zhi Wang, 1-Oct-2024.) |
⊢ 𝐵 = (Base‘𝐷) & ⊢ 𝐶 = (Base‘𝐸) & ⊢ 𝐻 = (Hom ‘𝐷) & ⊢ 𝐽 = (Hom ‘𝐸) & ⊢ (𝜑 → 𝐷 ∈ Cat) & ⊢ (𝜑 → 𝐸 ∈ ThinCat) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ 𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅)) ⇒ ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾)) | ||
Theorem | fullthinc 45943* | A functor to a thin category is full iff empty hom-sets are mapped to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐷 ∈ ThinCat) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) ⇒ ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))) | ||
Theorem | fullthinc2 45944 | A full functor to a thin category maps empty hom-sets to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.) |
⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐽 = (Hom ‘𝐷) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝐷 ∈ ThinCat) & ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)) | ||
Theorem | thincfth 45945 | A functor from a thin category is faithful. (Contributed by Zhi Wang, 1-Oct-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) ⇒ ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) | ||
Theorem | thincciso 45946* | Two thin categories are isomorphic iff the induced preorders are order-isomorphic. Example 3.26(2) of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Oct-2024.) |
⊢ 𝐶 = (CatCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝑅 = (Base‘𝑋) & ⊢ 𝑆 = (Base‘𝑌) & ⊢ 𝐻 = (Hom ‘𝑋) & ⊢ 𝐽 = (Hom ‘𝑌) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ThinCat) & ⊢ (𝜑 → 𝑌 ∈ ThinCat) ⇒ ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆))) | ||
Theorem | 0thincg 45947 | Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat) | ||
Theorem | 0thinc 45948 | The empty category (see 0cat 17146) is thin. (Contributed by Zhi Wang, 17-Sep-2024.) |
⊢ ∅ ∈ ThinCat | ||
Theorem | indthinc 45949* | An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are ∅. This is a special case of prsthinc 45951, where ≤ = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | indthincALT 45950* | An alternate proof for indthinc 45949 assuming more axioms including ax-pow 5243 and ax-un 7501. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | prsthinc 45951* | Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 45908 and catprs2 45909 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶)) & ⊢ (𝜑 → ∅ = (comp‘𝐶)) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐶 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | ||
Theorem | setcthin 45952* | A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (SetCat‘𝑈)) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥) ⇒ ⊢ (𝜑 → 𝐶 ∈ ThinCat) | ||
Theorem | setc2othin 45953 | The category (SetCat‘2o) is thin. A special case of setcthin 45952. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (SetCat‘2o) ∈ ThinCat | ||
Theorem | thincsect 45954 | In a thin category, one morphism is a section of another iff they are pointing towards each other. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)))) | ||
Theorem | thincsect2 45955 | In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹)) | ||
Theorem | thincinv 45956 | In a thin category, 𝐹 is an inverse of 𝐺 iff 𝐹 is a section of 𝐺 (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑆 = (Sect‘𝐶) & ⊢ 𝑁 = (Inv‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) | ||
Theorem | thinciso 45957 | In a thin category, 𝐹:𝑋⟶𝑌 is an isomorphism iff there is a morphism from 𝑌 to 𝑋. (Contributed by Zhi Wang, 25-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ 𝐼 = (Iso‘𝐶) & ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅)) | ||
Theorem | thinccic 45958 | In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024.) |
⊢ (𝜑 → 𝐶 ∈ ThinCat) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅))) | ||
Syntax | cprstc 45959 | Class function defining preordered sets as categories. |
class ProsetToCat | ||
Definition | df-prstc 45960 |
Definition of the function converting a preordered set to a category.
Justified by prsthinc 45951.
This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism, by prstcnid 45963, prstchom 45972, and prstcthin 45971. Other important properties include prstcbas 45964, prstcleval 45965, prstcle 45966, prstcocval 45967, prstcoc 45968, prstchom2 45973, and prstcprs 45970. Use those instead. Note that the defining property prstchom 45972 is equivalent to prstchom2 45973 given prstcthin 45971. See thincn0eu 45929 for justification. "ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet 〈(Hom ‘ndx), ((le‘𝑘) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) | ||
Theorem | prstcval 45961 | Lemma for prstcnidlem 45962 and prstcthin 45971. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 = ((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) | ||
Theorem | prstcnidlem 45962 | Lemma for prstcnid 45963 and prstchomval 45969. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (comp‘ndx) ⇒ ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) | ||
Theorem | prstcnid 45963 | Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (comp‘ndx) & ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx) ⇒ ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) | ||
Theorem | prstcbas 45964 | The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | ||
Theorem | prstcleval 45965 | Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐾)) ⇒ ⊢ (𝜑 → ≤ = (le‘𝐶)) | ||
Theorem | prstcle 45966 | Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐾)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐶)𝑌)) | ||
Theorem | prstcocval 45967 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ⊥ = (oc‘𝐾)) ⇒ ⊢ (𝜑 → ⊥ = (oc‘𝐶)) | ||
Theorem | prstcoc 45968 | Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ⊥ = (oc‘𝐾)) ⇒ ⊢ (𝜑 → ( ⊥ ‘𝑋) = ((oc‘𝐶)‘𝑋)) | ||
Theorem | prstchomval 45969 | Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) ⇒ ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶)) | ||
Theorem | prstcprs 45970 | The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 ∈ Proset ) | ||
Theorem | prstcthin 45971 | The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → 𝐶 ∈ ThinCat) | ||
Theorem | prstchom 45972 |
Hom-sets of the constructed category are dependent on the preorder.
Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) | ||
Theorem | prstchom2 45973* |
Hom-sets of the constructed category are dependent on the preorder.
Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat ( see prstchom2ALT 45974). However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 21-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | ||
Theorem | prstchom2ALT 45974* | Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 45960. See prstchom2 45973 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐶)) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) | ||
Theorem | postcpos 45975 | The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset)) | ||
Theorem | postcposALT 45976 | Alternate proof for postcpos 45975. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) ⇒ ⊢ (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset)) | ||
Theorem | postc 45977* | The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦))) | ||
Syntax | cmndtc 45978 | Class function defining monoids as categories. |
class MndToCat | ||
Definition | df-mndtc 45979 |
Definition of the function converting a monoid to a category. Example
3.3(4.e) of [Adamek] p. 24.
The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 45981) , instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24. The resulting category is defined entirely, up to isomorphism, by mndtcbas 45982, mndtchom 45985, mndtcco 45986. Use those instead. See example 3.26(3) of [Adamek] p. 33 for more on isomorphism. "MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ MndToCat = (𝑚 ∈ Mnd ↦ {〈(Base‘ndx), {𝑚}〉, 〈(Hom ‘ndx), {〈𝑚, 𝑚, (Base‘𝑚)〉}〉, 〈(comp‘ndx), {〈〈𝑚, 𝑚, 𝑚〉, (+g‘𝑚)〉}〉}) | ||
Theorem | mndtcval 45980 | Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), {𝑀}〉, 〈(Hom ‘ndx), {〈𝑀, 𝑀, (Base‘𝑀)〉}〉, 〈(comp‘ndx), {〈〈𝑀, 𝑀, 𝑀〉, (+g‘𝑀)〉}〉}) | ||
Theorem | mndtcbasval 45981 | The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = {𝑀}) | ||
Theorem | mndtcbas 45982* | The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) ⇒ ⊢ (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵) | ||
Theorem | mndtcob 45983 | Lemma for mndtchom 45985 and mndtcco 45986. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 = 𝑀) | ||
Theorem | mndtcbas2 45984 | Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
Theorem | mndtchom 45985 | The only hom-set of the category built from a monoid is the base set of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (Base‘𝑀)) | ||
Theorem | mndtcco 45986 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → · = (comp‘𝐶)) ⇒ ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (+g‘𝑀)) | ||
Theorem | mndtcco2 45987 | The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → · = (comp‘𝐶)) & ⊢ (𝜑 → ⚬ = (〈𝑋, 𝑌〉 · 𝑍)) ⇒ ⊢ (𝜑 → (𝐺 ⚬ 𝐹) = (𝐺(+g‘𝑀)𝐹)) | ||
Theorem | mndtccatid 45988* | Lemma for mndtccat 45989 and mndtcid 45990. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀)))) | ||
Theorem | mndtccat 45989 | The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) ⇒ ⊢ (𝜑 → 𝐶 ∈ Cat) | ||
Theorem | mndtcid 45990 | The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 1 = (Id‘𝐶)) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = (0g‘𝑀)) | ||
Theorem | grptcmon 45991 | All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝑀 = (Mono‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌)) | ||
Theorem | grptcepi 45992 | All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.) |
⊢ (𝜑 → 𝐶 = (MndToCat‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) & ⊢ (𝜑 → 𝐸 = (Epi‘𝐶)) ⇒ ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌)) | ||
Some of these theorems are used in the series of lemmas and theorems proving the defining properties of setrecs. | ||
Theorem | nfintd 45993 | Bound-variable hypothesis builder for intersection. (Contributed by Emmett Weisz, 16-Jan-2020.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∩ 𝐴) | ||
Theorem | nfiund 45994* | Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2371. See nfiundg 45995 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝐴) & ⊢ (𝜑 → Ⅎ𝑦𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | nfiundg 45995 | Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2371, see nfiund 45994 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝐴) & ⊢ (𝜑 → Ⅎ𝑦𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunord 45996* | The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7541, but does not use it directly, since ssorduni 7541 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.) |
⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunordi 45997* | The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. (Contributed by Emmett Weisz, 3-Nov-2019.) |
⊢ Ord 𝐵 ⇒ ⊢ Ord ∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | spd 45998 | Specialization deduction, using implicit substitution. Based on the proof of spimed 2387. (Contributed by Emmett Weisz, 17-Jan-2020.) |
⊢ (𝜒 → Ⅎ𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spcdvw 45999* | A version of spcdv 3499 where 𝜓 and 𝜒 are direct substitutions of each other. This theorem is useful because it does not require 𝜑 and 𝑥 to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | tfis2d 46000* | Transfinite Induction Schema, using implicit substitution. (Contributed by Emmett Weisz, 3-May-2020.) |
⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) & ⊢ (𝜑 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜒 → 𝜓))) ⇒ ⊢ (𝜑 → (𝑥 ∈ On → 𝜓)) |
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