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Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoppchomf 13901 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       f        tpos f

Theoremoppcid 13902 Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccat 13903 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theorem2oppcbas 13904 The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 13918. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              oppCat

Theorem2oppchomf 13905 The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 13918. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat       f f oppCat

Theorem2oppccomf 13906 The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 13918. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat       compf compfoppCat

Theoremoppchomfpropd 13907 If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        f oppCat f oppCat

Theoremoppccomfpropd 13908 If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf       compfoppCat compfoppCat

8.1.3  Monomorphisms and epimorphisms

Syntaxcmon 13909 Extend class notation with the class of all monomorphisms.
Mono

Syntaxcepi 13910 Extend class notation with the class of all epimorphisms.
Epi

Definitiondf-mon 13911* Function returning the monomorphisms of the category . JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
Mono comp

Definitiondf-epi 13912 Function returning the epimorphisms of the category . JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Epi tpos MonooppCat

Theoremmonfval 13913* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp       Mono

Theoremismon 13914* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremismon2 13915* Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmonhom 13916 A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmoni 13917 Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmonpropd 13918 If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
f f        compf compf                     Mono Mono

Theoremoppcmon 13919 A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              Mono       Epi

Theoremoppcepi 13920 An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              Epi       Mono

Theoremisepi 13921* Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremisepi2 13922* Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremepihom 13923 An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremepii 13924 Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp       Epi

8.1.4  Sections, inverses, isomorphisms

Syntaxcsect 13925 Extend class notation with the sections of a morphism.
Sect

Syntaxcinv 13926 Extend class notation with the inverses of a morphism.
Inv

Syntaxciso 13927 Extend class notation with the class of all isomorphisms.

Definitiondf-sect 13928* Function returning the section relation in a category. Given arrows and , we say Sect, that is, is a section of , if . (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect comp

Definitiondf-inv 13929* The inverse relation in a category. Given arrows and , we say Inv, that is, is an inverse of , if is a section of and is a section of . (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Inv Sect Sect

Definitiondf-iso 13930* Function returning the isomorphisms of the category . The Joy of Cats p. 28. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 2-Jan-2017.)
Inv

Theoremsectffval 13931* Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectfval 13932* Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectss 13933 The section relation is a relation between morphisms from to and morphisms from to . (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremissect 13934 The property " is a section of ". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremissect2 13935 Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectcan 13936 If is a section of and is a section of , then . Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect

Theoremsectco 13937 Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Sect

Theoreminvffval 13938* Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoreminvfval 13939 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoremisinv 13940 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoreminvss 13941 The inverse relation is a relation between morphisms and their inverses . (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvsym 13942 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvsym2 13943 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvfun 13944 The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoremisoval 13945 The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminviso1 13946 If is an inverse to , then is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoreminviso2 13947 If is an inverse to , then is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoreminvf 13948 The inverse relation is a function from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvf1o 13949 The inverse relation is a bijection from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvinv 13950 The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvco 13951 The composition of two isomorphisms is an isomorphism, and the inverse is the composition of the individual inverses. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                                          comp

Theoremisohom 13952 An isomorphism is a homomorphism. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremisoco 13953 The composition of two isomorphisms is an isomorphism. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremoppcsect 13954 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Sect       Sect

Theoremoppcsect2 13955 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Sect       Sect

Theoremoppcinv 13956 An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Inv       Inv

Theoremoppciso 13957 An isomorphism in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat

Theoremsectmon 13958 If is a section of , then is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono       Sect

Theoremmonsect 13959 If is a monomorphism and is a section of , then is an inverse of and they are both isomorphisms. This is also stated as "a monomorphism which is also a split epimorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono       Sect                            Inv

Theoremsectepi 13960 If is a section of , then is an epimorphism. An epimorphism that arises from a section is also known as a split epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi       Sect

Theoremepisect 13961 If is an epimorphism and is a section of , then is an inverse of and they are both isomorphisms. This is also stated as "an epimorphism which is also a split monomorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi       Sect                            Inv

8.1.5  Subcategories

Syntaxcssc 13962 Extend class notation to include the subset relation for subcategories.
cat

Syntaxcresc 13963 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
cat

Syntaxcsubc 13964 Extend class notation to include the collection of subcategories of a category.
Subcat

Definitiondf-ssc 13965* Define the subset relation for subcategories. Despite the name, this is not really a "category-aware" definition, which is to say it makes no explicit references to homsets or composition; instead this is a subset-like relation on the functions that are used as subcategory specifications in df-subc 13967, which makes it play an analogous role to the subset relation applied to the subgroups of a group. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Definitiondf-resc 13966* Define the restriction of a category to a given set of arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat s sSet

Definitiondf-subc 13967* Subcat is the set of all the subcategory specifications of the category . Like df-subg 14896, this is not actually a collection of categories, but only sets which when given operations from the base category (using df-resc 13966) form a category. All the objects and all the morphisms of the subcategory belong to the supercategory. The identity of an object, the domain and the codomain of a morphism are the same in the subcategory and the supercategory. The composition of the subcategory is a restriction of the composition of the supercategory. (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 4-Jan-2017.)
Subcat cat f comp

Theoremsscrel 13968 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theorembrssc 13969* The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscpwex 13970* An analogue of pwex 4342 for the subcategory subset relation: The collection of subcategory subsets of a given set is a set. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsubcrcl 13971 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat

Theoremsscfn1 13972 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscfn2 13973 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssclem 13974 Lemma for ssc1 13976 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremisssc 13975* Value of the subcategory subset relation when the arguments are known functions. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssc1 13976 Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssc2 13977 Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscres 13978 Any function restricted to a square domain is a subcategory subset of the original. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscid 13979 The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssctr 13980 The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat cat

Theoremssceq 13981 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat

Theoremrescval 13982 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        s sSet

Theoremrescval2 13983 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                             s sSet

Theoremrescbas 13984 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat

Theoremreschom 13985 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat

Theoremreschomf 13986 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                                    f

Theoremrescco 13987 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                                    comp       comp

Theoremrescabs 13988 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat cat

Theoremrescabs2 13989 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
s cat cat

Theoremissubc 13990* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
f               comp                     Subcat cat

Theoremissubc2 13991* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
f               comp                     Subcat cat

Theoremsubcssc 13992 An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat       f        cat

Theoremsubcfn 13993 An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcss1 13994 The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcss2 13995 The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcidcl 13996 The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubccocl 13997 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat                     comp

Theoremsubccatid 13998* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

Theoremsubcid 13999 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

Theoremsubccat 14000 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

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