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Theorem List for Metamath Proof Explorer - 13501-13600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremissect2 13501 Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectcan 13502 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 13503 Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Sect

Theoreminvffval 13504* Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoreminvfval 13505 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoremisinv 13506 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoreminvss 13507 The inverse relation is a relation between morphisms and their inverses . (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvsym 13508 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvsym2 13509 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvfun 13510 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 13511 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 13512 If is an inverse to , then is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoreminviso2 13513 If is an inverse to , then is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoreminvf 13514 The inverse relation is a function from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvf1o 13515 The inverse relation is a bijection from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvinv 13516 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 13517 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 13518 An isomorphism is a homomorphism. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremisoco 13519 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 13520 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Sect       Sect

Theoremoppcsect2 13521 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Sect       Sect

Theoremoppcinv 13522 An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Inv       Inv

Theoremoppciso 13523 An isomorphism in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat

Theoremsectmon 13524 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 13525 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 13526 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 13527 If is an epimorphism and is a section of , then is an inverse of and they are both isomorphisms. This is also stated as "a 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 13528 Extend class notation to include the subset relation for subcategories.
cat

Syntaxcresc 13529 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
cat

Syntaxcsubc 13530 Extend class notation to include the collection of subcategories of a category.
Subcat

Definitiondf-ssc 13531* 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 13533, 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 13532* Define the restriction of a category to a given set of arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat s sSet

Definitiondf-subc 13533* Subcat is the set of all the subcategory specifications of the category . Like df-subg 14453, this is not actually a collection of categories, but only sets which when given operations from the base category (using df-resc 13532) 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 13534 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theorembrssc 13535* The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscpwex 13536* An analogue of pwex 4087 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 13537 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat

Theoremsscfn1 13538 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscfn2 13539 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssclem 13540 Lemma for ssc1 13542 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremisssc 13541* Value of the subcategory subset relation when the arguments are known functions. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssc1 13542 Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssc2 13543 Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscres 13544 Any function restricted to a square domain is a subcategory subset of the original. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscid 13545 The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssctr 13546 The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat cat

Theoremssceq 13547 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat

Theoremrescval 13548 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        s sSet

Theoremrescval2 13549 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                             s sSet

Theoremrescbas 13550 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat

Theoremreschom 13551 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat

Theoremreschomf 13552 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                                    f

Theoremrescco 13553 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                                    comp       comp

Theoremrescabs 13554 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat cat

Theoremrescabs2 13555 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
s cat cat

Theoremissubc 13556* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
f               comp                     Subcat cat

Theoremissubc2 13557* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
f               comp                     Subcat cat

Theoremsubcssc 13558 An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat       f        cat

Theoremsubcfn 13559 An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcss1 13560 The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcss2 13561 The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcidcl 13562 The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubccocl 13563 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat                     comp

Theoremsubccatid 13564* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

Theoremsubcid 13565 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

Theoremsubccat 13566 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

Theoremissubc3 13567* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 14261, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
f               cat                      Subcat cat

Theoremfullsubc 13568 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory). (Contributed by Mario Carneiro, 4-Jan-2017.)
f                      Subcat

Theoremfullresc 13569 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
f                      s        cat        f f compf compf

Theoremresscat 13570 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
s

Theoremsubsubc 13571 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat        Subcat Subcat Subcat cat

8.1.6  Functors

Syntaxcfunc 13572 Extend class notation with the class of all functors.

Syntaxcidfu 13573 Extend class notation with identity functor.
idfunc

Syntaxccofu 13574 Extend class notation with functor composition.
func

Syntaxcresf 13575 Extend class notation to include restriction of a functor to a subcategory.
f

Definitiondf-func 13576* Function returning all the functors from a category to a category . Intuitively a functor associates any morphism of to a morphism of , any object of to an object of , and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of to an object of we write it associates any identity of to an identity of which simplifies the definition. (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
comp comp

Definitiondf-idfu 13577* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Definitiondf-cofu 13578* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Definitiondf-resf 13579* Define the restriction of a functor to a subcategory (analogue of df-res 4600). (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremrelfunc 13580 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncrcl 13581 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremisfunc 13582* Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       comp

Theoremisfuncd 13583* Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp

Theoremfuncf1 13584 The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncixp 13585* The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncf2 13586 The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncfn2 13587 The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremfuncid 13588 A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncco 13589 A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       comp

Theoremfuncsect 13590 The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect       Sect

Theoremfuncinv 13591 The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv       Inv

Theoremfunciso 13592 The image of an isomorphism under a functor is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremfuncoppc 13593 A functor on categories yields a functor on the opposite categories (in the same direction). (Contributed by Mario Carneiro, 4-Jan-2017.)
oppCat       oppCat              tpos

Theoremidfuval 13594* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu2nd 13595 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu2 13596 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
idfunc

Theoremidfu1st 13597 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu1 13598 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfucl 13599 The identity functor is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremcofuval 13600* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

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