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

Definitiondf-comf 13901* Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf comp

Theoremiscat 13902* The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremiscatd 13903* Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatidex 13904* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatideu 13905* Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcidfval 13906* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremcidval 13907* Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcidffn 13908 The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)

Theoremcidfn 13909 The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremcatidd 13910* Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremiscatd2 13911* Version of iscatd2 13911 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp

Theoremcatidcl 13912 Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcatlid 13913 Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatrid 13914 Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatcocl 13915 Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremcatass 13916 Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theorem0catg 13917 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theorem0cat 13918 The empty set is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremproplem2 13919* Lemma for mndpropd 14726. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem 13920* Lemma for mndpropd 14726. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem3 13921 Lemma for property theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)

Theoremhomffval 13922* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfval 13923 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomffn 13924 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfeq 13925* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
f f

Theoremhomfeqd 13926 If two structures have the same slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqbas 13927 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqval 13928 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremcomfffval 13929* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomffval 13930* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomfval 13931 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomfffval2 13932* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomffval2 13933* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomfval2 13934 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomfffn 13935 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf

Theoremcomffn 13936 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf

Theoremcomfeq 13937* Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp                            f f        compf compf

Theoremcomfeqd 13938 Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp comp       f f        compf compf

Theoremcomfeqval 13939 Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp       f f        compf compf

Theoremcatpropd 13940 Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
f f        compf compf

Theoremcidpropd 13941 Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
f f        compf compf

8.1.2  Opposite category

Syntaxcoppc 13942 The opposite category operation.
oppCat

Definitiondf-oppc 13943* Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat sSet tpos sSet comp tpos comp

Theoremoppcval 13944* Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat       sSet tpos sSet comp tpos

Theoremoppchomfval 13945 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat       tpos

Theoremoppchom 13946 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccofval 13947 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat                            comp tpos

Theoremoppcco 13948 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat                            comp

Theoremoppcbas 13949 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccatid 13950 Lemma for oppccat 13953. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat

Theoremoppchomf 13951 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       f        tpos f

Theoremoppcid 13952 Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccat 13953 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theorem2oppcbas 13954 The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 13968. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              oppCat

Theorem2oppchomf 13955 The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 13968. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat       f f oppCat

Theorem2oppccomf 13956 The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 13968. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat       compf compfoppCat

Theoremoppchomfpropd 13957 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 13958 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 13959 Extend class notation with the class of all monomorphisms.
Mono

Syntaxcepi 13960 Extend class notation with the class of all epimorphisms.
Epi

Definitiondf-mon 13961* 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 13962 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 13963* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp       Mono

Theoremismon 13964* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremismon2 13965* Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmonhom 13966 A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmoni 13967 Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmonpropd 13968 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 13969 A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              Mono       Epi

Theoremoppcepi 13970 An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              Epi       Mono

Theoremisepi 13971* Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremisepi2 13972* Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremepihom 13973 An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremepii 13974 Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp       Epi

8.1.4  Sections, inverses, isomorphisms

Syntaxcsect 13975 Extend class notation with the sections of a morphism.
Sect

Syntaxcinv 13976 Extend class notation with the inverses of a morphism.
Inv

Syntaxciso 13977 Extend class notation with the class of all isomorphisms.

Definitiondf-sect 13978* 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 13979* 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 13980* 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 13981* Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectfval 13982* Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectss 13983 The section relation is a relation between morphisms from to and morphisms from to . (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremissect 13984 The property " is a section of ". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremissect2 13985 Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectcan 13986 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 13987 Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Sect

Theoreminvffval 13988* Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoreminvfval 13989 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoremisinv 13990 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoreminvss 13991 The inverse relation is a relation between morphisms and their inverses . (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvsym 13992 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvsym2 13993 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvfun 13994 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 13995 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 13996 If is an inverse to , then is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoreminviso2 13997 If is an inverse to , then is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoreminvf 13998 The inverse relation is a function from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvf1o 13999 The inverse relation is a bijection from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvinv 14000 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

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