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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-coda 13701 Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
coda

Definitiondf-homa 13702* 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-dom_ 24883 and df-cod_ 24884. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Homa

Definitiondf-arw 13703 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 13704 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomafval 13705* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomaf 13706 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomaval 13707 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhoma 13708 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhomai 13709 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhomai2 13710 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomarcl2 13711 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomarel 13712 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhoma1 13713 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomahom2 13714 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomahom 13715 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomadm 13716 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomacd 13717 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       coda

Theoremhomadmcd 13718 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremarwval 13719 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa

Theoremarwrcl 13720 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremarwhoma 13721 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 13722 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa

Theoremarwdm 13723 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremarwcd 13724 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremdmaf 13725 The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremcdaf 13726 The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremarwhom 13727 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 13728 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 13729 Extend class notation to include identity for arrows.
Ida

Syntaxccoa 13730 Extend class notation to include composition for arrows.
compa

Definitiondf-ida 13731* 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 13732* 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 13733* Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidaval 13734 Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremida2 13735 Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidahom 13736 Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            Homa

Theoremidadm 13737 Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidacd 13738 Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            coda

Theoremidaf 13739 The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                     Nat

Theoremcoafval 13740* The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat       comp       coda coda coda

Theoremeldmcoa 13741 A pair is in the domain of the arrow composition, if the domain of equals the codomain of . (In this case we say and are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat       coda

Theoremdmcoass 13742 The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat

Theoremhomdmcoa 13743 If and , then and are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa

Theoremcoaval 13744 Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa                     comp

Theoremcoa2 13745 The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa                     comp

Theoremcoahom 13746 The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa

Theoremcoapm 13747 Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat

Theoremarwlid 13748 Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

Theoremarwrid 13749 Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

Theoremarwass 13750 Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

8.3  Examples of categories

8.3.1  The category of sets

Syntaxcsetc 13751 Extend class notation to include the category Set.

Definitiondf-setc 13752* Definition of the category Set, relativized to a subset . This is the category of all sets in and functions between these sets. Generally, we will take to be a weak universe or Grothendieck's universe, because these sets have closure properties as good as the real thing. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcval 13753* Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcbas 13754 Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetchomfval 13755* Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetchom 13756 Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremelsetchom 13757 A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetccofval 13758* Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcco 13759 Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetccatid 13760* Lemma for setccat 13761. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetccat 13761 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetcid 13762 The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetcmon 13763 A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono

Theoremsetcepi 13764 A epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi

Theoremsetcsect 13765 A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
Sect

Theoremsetcinv 13766 An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoremsetciso 13767 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremresssetc 13768 The restriction of the category of sets to a subset is the category of sets in the subset. Thus the categories for different are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
f s f compfs compf

Theoremfuncsetcres2 13769 A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)

8.3.2  The category of categories

Syntaxccatc 13770 Extend class notation to include the category Cat.
CatCat

Definitiondf-catc 13771* Definition of the category Cat, which consists of all categories in the universe , with functors as the morphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat comp func

Theoremcatcval 13772* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                            func        comp

Theoremcatcbas 13773 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchomfval 13774* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchom 13775 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatccofval 13776* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp       func

Theoremcatcco 13777 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp                                          func

Theoremcatccatid 13778* Lemma for catccat 13780. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat              idfunc

Theoremcatcid 13779 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     idfunc

Theoremcatccat 13780 The category of categories is a category. (Clearly it cannot be an element of itself, hence it is "large" with respect to .) (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremresscatc 13781 The restriction of the category of categories to a subset is the category of categories in the subset. Thus the CatCat categories for different are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
CatCat       CatCat                     f s f compfs compf

Theoremcatcisolem 13782* Lemma for catciso 13783. (Contributed by Mario Carneiro, 29-Jan-2017.)
CatCat                                                 Inv              Full Faith

Theoremcatciso 13783 A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. (Contributed by Mario Carneiro, 29-Jan-2017.)
CatCat                                                        Full Faith

Theoremcatcoppccl 13784 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              oppCat       WUni

Theoremcatcfuccl 13785 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              FuncCat        WUni

8.4  Categorical constructions

8.4.1  Product of categories

Syntaxcxpc 13786 Extend class notation with the product of two categories.
c

Syntaxc1stf 13787 Extend class notation with the first projection functor.
F

Syntaxc2ndf 13788 Extend class notation with the second projection functor.
F

Syntaxcprf 13789 Extend class notation with the functor pairing operation.
⟨,⟩F

Definitiondf-xpc 13790* Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)
c comp comp comp

Definitiondf-1stf 13791* Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
F c

Definitiondf-2ndf 13792* Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
F c

Definitiondf-prf 13793* Define the pairing operation for functors (which takes two functors and and produces ⟨,⟩F c ). (Contributed by Mario Carneiro, 11-Jan-2017.)
⟨,⟩F

Theoremfnxpc 13794 The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
c

Theoremxpcval 13795* Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
c                                    comp       comp                                          comp

Theoremxpcbas 13796 Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
c

Theoremxpchomfval 13797* Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpchom 13798 Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremrelxpchom 13799 A hom-set in the binary product of categories is a relation. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpccofval 13800* Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp       comp       comp

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