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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfthpropd 14101 If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
f f        compf compf       f f        compf compf                                   Faith Faith

Theoremfulloppc 14102 The opposite functor of a full functor is also full. (Contributed by Mario Carneiro, 27-Jan-2017.)
oppCat       oppCat       Full        Full tpos

Theoremfthoppc 14103 The opposite functor of a faithful functor is also faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
oppCat       oppCat       Faith        Faith tpos

Theoremffthoppc 14104 The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
oppCat       oppCat       Full Faith        Full Faith tpos

Theoremfthsect 14105 A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                                    Sect       Sect

Theoremfthinv 14106 A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                                    Inv       Inv

Theoremfthmon 14107 A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Mono       Mono

Theoremfthepi 14108 A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Epi       Epi

Theoremffthiso 14109 A fully faithful functor reflects isomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Faith                             Full

Theoremfthres2b 14110* Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
Subcat                            Faith Faith cat

Theoremfthres2c 14111 Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
s                             Faith Faith

Theoremfthres2 14112 A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 27-Jan-2017.)
Subcat Faith cat Faith

Theoremidffth 14113 The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
idfunc       Full Faith

Theoremcofull 14114 The composition of two full functors is full. (Contributed by Mario Carneiro, 28-Jan-2017.)
Full        Full        func Full

Theoremcofth 14115 The composition of two faithful functors is faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Faith        Faith        func Faith

Theoremcoffth 14116 The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
Full Faith        Full Faith        func Full Faith

Theoremrescfth 14117 The inclusion functor from a subcategory is a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
cat        idfunc       Subcat Faith

Theoremressffth 14118 The inclusion functor from a full subcategory is a full and faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
s        idfunc       Full Faith

Theoremfullres2c 14119 Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
s                             Full Full

Theoremffthres2c 14120 Condition for a fully faithful functor to also be a fully faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
s                             Full Faith Full Faith

8.1.8  Natural transformations and the functor category

Syntaxcnat 14121 Extend class notation to include the collection of natural transformations.
Nat

Syntaxcfuc 14122 Extend class notation to include the functor category.
FuncCat

Definitiondf-nat 14123* Definition of a natural transformation between two functors. A natural transformation is a collection of arrows , such that for each morphism . (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat comp comp

Definitiondf-fuc 14124* Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat Nat comp Nat Nat comp

Theoremfnfuc 14125 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
FuncCat

Theoremnatfval 14126* Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremisnat 14127* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremisnat2 14128* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremnatffn 14129 The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
Nat

Theoremnatrcl 14130 Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnat1st2nd 14131 Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnatixp 14132* A natural transformation is a function from the objects of to homomorphisms from to . (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnatcl 14133 A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnatfn 14134 A natural transformation is a function on the objects of . (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat

Theoremnati 14135 Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Nat                             comp

Theoremwunnat 14136 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
WUni                     Nat

Theoremcatstr 14137 A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp Struct ;

Theoremfucval 14138* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat               Nat               comp                            comp

Theoremfuccofval 14139* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat               Nat               comp                     comp

Theoremfucbas 14140 The objects of the functor category are functors from to . (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.)
FuncCat

Theoremfuchom 14141 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat

Theoremfucco 14142* Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat               comp       comp

Theoremfuccoval 14143 Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat               comp       comp

Theoremfuccocl 14144 The composition of two natural transformations is a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucidcl 14145 The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat

Theoremfuclid 14146 Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucrid 14147 Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucass 14148 Associativity of natural transformation composition. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfuccatid 14149* The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfuccat 14150 The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfucid 14151 The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfucsect 14152* Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Sect       Sect

Theoremfucinv 14153* Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Inv       Inv

Theoreminvfuc 14154* If is an inverse to for each , and is a natural transformation, then is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Inv       Inv

Theoremfuciso 14155* A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat

Theoremnatpropd 14156 If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf       f f        compf compf                                   Nat Nat

Theoremfucpropd 14157 If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf       f f        compf compf                                   FuncCat FuncCat

8.2  Arrows (disjointified hom-sets)

Syntaxcdoma 14158 Extend class notation to include the domain extractor for an arrow.

Syntaxccoda 14159 Extend class notation to include the codomain extractor for an arrow.
coda

Syntaxcarw 14160 Extend class notation to include the collection of all arrows of a category.
Nat

Syntaxchoma 14161 Extend class notation to include the set of all arrows with a specific domain and codomain.
Homa

Definitiondf-doma 14162 Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)

Definitiondf-coda 14163 Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
coda

Definitiondf-homa 14164* Definition of the hom-set extractor for arrows, which tags the morphisms of the underlying hom-set with domain and codomain, which can then be extracted using df-doma 14162 and df-coda 14163. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Homa

Definitiondf-arw 14165 Definition of the set of arrows of a category. We will use the term "arrow" to denote a morphism tagged with its domain and codomain, as opposed to , which allows hom-sets for distinct objects to overlap. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat Homa

Theoremhomarcl 14166 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomafval 14167* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomaf 14168 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomaval 14169 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhoma 14170 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhomai 14171 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhomai2 14172 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomarcl2 14173 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomarel 14174 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhoma1 14175 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomahom2 14176 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomahom 14177 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomadm 14178 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomacd 14179 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       coda

Theoremhomadmcd 14180 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremarwval 14181 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa

Theoremarwrcl 14182 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremarwhoma 14183 An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa       coda

Theoremhomarw 14184 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa

Theoremarwdm 14185 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremarwcd 14186 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremdmaf 14187 The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremcdaf 14188 The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremarwhom 14189 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremarwdmcd 14190 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       coda

8.2.1  Identity and composition for arrows

Syntaxcida 14191 Extend class notation to include identity for arrows.
Ida

Syntaxccoa 14192 Extend class notation to include composition for arrows.
compa

Definitiondf-ida 14193* Definition of the identity arrow, which is just the identity morphism tagged with its domain and codomain. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Ida

Definitiondf-coa 14194* Definition of the composition of arrows. Since arrows are tagged with domain and codomain, this does not need to be a 5-ary operation like the regular composition in a category comp. Instead, it is a partial binary operation on arrows, which is defined when the domain of the first arrow matches the codomain of the second. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa Nat Nat coda coda compcoda

Theoremidafval 14195* Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidaval 14196 Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremida2 14197 Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidahom 14198 Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            Homa

Theoremidadm 14199 Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidacd 14200 Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            coda

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