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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremishomc 25201 The predicate JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)

Theoremishomd 25202 The predicate JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)

Theoremehm 25203 The elements of a homset are morphisms. JFM CAT1 th. 21. (Contributed by FL, 5-Dec-2007.)

Theoremdehm 25204 Domain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)

Theoremcehm 25205 Codomain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)

Theoremmrdmcd 25206 A morphism belongs to the homset between its domain and its codomain. JFM CAT1 th. 22. (Contributed by FL, 1-Nov-2007.)

Theoremeqidob 25207 When the identities are equal, the objects are equal. JFM CAT1 th. 45. (Contributed by FL, 24-Apr-2007.)

Theoremhomib 25208 The homset which belongs to. JFM CAT1 th. 55. (Contributed by FL, 5-Dec-2007.)

Theoremhine 25209 The homset is not empty. JFM CAT1 th. 56. (Contributed by FL, 3-Jan-2008.)

Theoremcmphmia 25210 Composite of the member of a homset with the identity. JFM CAT1 th. 57 (Contributed by FL, 5-Dec-2007.)

Theoremcmphmib 25211 Composite of a member of a homset with the identity. JFM CAT1 th. 58 (Contributed by FL, 5-Dec-2007.)

Theoremiri 25212 Composite of an identity with itself. JFM CAT1 th. 59 (Contributed by FL, 5-Dec-2007.)

Theoremcmpassoh 25213 is associative. Homset-based version of cmpasso 25185. (Contributed by FL, 10-Mar-2008.)

Theoremhomgrf 25214 Homset of a composite. JFM CAT1 th. 51 (Contributed by FL, 10-Mar-2008.)

18.13.50  Monomorphisms, Epimorphisms, Isomorphisms

SyntaxcepiOLD 25215 Extend class notation with the class of all epimorphisms.
Epic

SyntaxcmonOLD 25216 Extend class notation with the class of all monomorphisms.
MonoOLD

SyntaxcisoOLD 25217 Extend class notation with the class of all isomorphisms.

Definitiondf-monOLD 25218* Function returning the monomorphisms of the category . JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.)
MonoOLD

Definitiondf-epiOLD 25219* Function returning the epimorphisms of the category . JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.)
Epic

Definitiondf-isoc 25220* Function returning the isomorphisms of the category . The Joy of Cats p. 28. (Contributed by FL, 9-Jun-2014.)

Theoremismona 25221* Monomorphisms of a category. (Contributed by FL, 5-Dec-2007.)
MonoOLD

Theoremismonb 25222* The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremismonb1 25223* The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremismonb2 25224 A monomorphism is a left-cancelable morphism. (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremimonclem 25225* Lemma for ismonc 25226. (Contributed by FL, 1-Jan-2008.)

Theoremismonc 25226* The predicate "is a monomorphism" when the morphism belongs to a homset. (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremcmpmon 25227 The composite of two monomorphisms is a monomorphism. JFM CAT1 th. 61 (Contributed by FL, 10-Mar-2008.)
MonoOLD MonoOLD MonoOLD

Theoremicmpmon 25228 If is a monomorphism then is a monomorphism. JFM CAT1 th. 62 (Contributed by FL, 17-Mar-2008.)
MonoOLD MonoOLD

Theoremidmon 25229 If there exists such as then F is a monomorphism. JFM CAT1 th. 63. (Contributed by FL, 5-May-2008.)
MonoOLD

Theoremimmon 25230 A morphism identity is a monomorphism. JFM CAT1 th. 64. (Contributed by FL, 5-May-2008.)
MonoOLD

Theoremisepia 25231* Epimorphisms of a category . (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
Epic

Theoremisepib 25232* The predicate "is an epimorphism". (Contributed by FL, 8-Aug-2008.)
Epic

Theoremisepib1 25233* The predicate "is an epimorphism". (Contributed by FL, 10-Aug-2008.)
Epic

Theoremisepib2 25234 An epimorphism is a right-cancelable morphism. (Contributed by FL, 10-Aug-2008.)
Epic

Theoremiepiclem 25235* Lemma for isepic 25236. (Contributed by FL, 6-Oct-2008.)

Theoremisepic 25236* The predicate "is an epimorphism" when the morphism belongs to a homset. (Contributed by FL, 27-Oct-2008.)
Epic

Theoremisiso 25237* Isomorphisms of a category. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 11-Sep-2015.)

Syntaxccinv 25238 Extend class notation to include a function that returns the inverses of a morphism.

Definitiondf-cinv 25239* Function ( indexed by the category ) returning the inverses of a morphism . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem1 25240* The set of the inverses of all the morphisms . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem2 25241* The set of the inverses of the morphism . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem3 25242 The set of the inverses of the morphism . (Contributed by FL, 22-Dec-2008.)

18.13.51  Functors

SyntaxcfuncOLD 25243 Extend class notation with the class of all functors.

Syntaxcifunc 25244 Extend class notation with the class of all isomorphisms.

Definitiondf-funcOLD 25245* 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.)

Theoremisfuna 25246* The class of functors between the categories and . (Contributed by FL, 10-Feb-2008.)

Theoremisfunb 25247* The predicate "is a functor" . (Contributed by FL, 10-Feb-2008.)

Theoremfmamo 25248 A functor is a mapping between morphisms. (Contributed by FL, 10-Feb-2008.)

Theoremvidfunid 25249* The functor associates every object of to an object in . For the identification of objects with the identities see df-funcOLD 25245. JFM CAT1 th. 97. (Contributed by FL, 10-Feb-2008.)

Theoremiddvvidd 25250* Functors preserve domains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)

Theoremidcvvidc 25251* Functors preserve codomains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)

Theoremfmp 25252* Functors preserve morphisms composition. JFM CAT1 th. 99. (Contributed by FL, 2-Aug-2009.)

Theoremidfisf 25253 The identity functor is a functor. (Contributed by FL, 15-Jul-2008.)

Definitiondf-isof 25254* Class of isomorphisms. (Contributed by FL, 21-May-2012.)

18.13.52  Subcategories

Syntaxcsubcat 25255 Extend class notation with a function returning all the subcategories of a given category.

Definitiondf-subcat 25256 is the set of all the subcategories of the 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.)

Theoremissubcat 25257 The set of all the subcategories of . (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremissubcata 25258 The predicate "is a subcategory of" . (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremissubcatb 25259 The predicate "is a subcategory of" . (Contributed by FL, 17-Sep-2009.)

Theorembesubbeca 25260 Lemma to simplify some subcategories related theorems . (Contributed by FL, 17-Sep-2009.)

Theoremcatsbc 25261 A category belongs to the set of its subcategories. (Contributed by FL, 17-Sep-2009.)

Theoremobsubc2 25262 The objects of a subcategory are a subset of those of the supercategory. JFM CAT2 th. 11 . (Contributed by FL, 17-Sep-2009.)

Theoremidsubc 25263 The identity function of a subcategory is a subset of the identity function of the supercategory. (Contributed by FL, 17-Sep-2009.)

Theoremdomsubc 25264 The domain function of a subcategory is a subset of the domain function of the supercategory. (Contributed by FL, 19-Sep-2009.)

Theoremcodsubc 25265 The codomain function of a subcategory is a subset of the codomain function of the supercategory. (Contributed by FL, 19-Sep-2009.)

Theoremsubctct 25266 A subcategory is a category. (Contributed by FL, 17-Sep-2009.)

Theoremmorsubc 25267 The morphisms of a subcategory are a subset of those of the supercategory. (Contributed by FL, 18-Sep-2009.)

Theoremcmpsubc 25268 The composition law of a subcategory is a subset of the composition law of the supercategory. (Contributed by FL, 20-Sep-2009.)

Theoremidsubidsup 25269* The identity of an an objet of the subcategory equals the identity of the object in the supercategory. (Contributed by FL, 2-Nov-2009.)

Theoremidsubfun 25270 The identity restricted to the morphism of a subcategory is a functor from the subcategory to the supercategory. It is called the inclusion functor. JFM CAT2 th. 19. (Contributed by FL, 5-Oct-2009.)

Theoreminfemb 25271 The inclusion functor is an embedding. (Contributed by FL, 2-Nov-2009.)

18.13.53  Terminal and initial objects

Syntaxciobj 25272 Extend class notation with the class of all initial objects.

Definitiondf-inob 25273* Definition of the initial objects of a category. Experimental. (Contributed by FL, 27-Jun-2014.)

Theoremisinob 25274* The predicate "are the initial objects of a category". (Contributed by FL, 27-Jun-2014.)

Syntaxctobj 25275 Extend class notation with the class of all terminal objects.

Definitiondf-termob 25276* Definition of the terminal objects of a category. Experimental. (Contributed by FL, 15-Jul-2012.)

18.13.54  Sources and sinks

Syntaxcsrce 25277 Extend class notation with the class of all sources.

Definitiondf-source 25278* A source is a family of morphims indexed by a set which all have the same domain . Joy of Cats, def. 10.1, p. 169. Experimental. (Contributed by FL, 30-May-2014.)

Theoremissrc 25279* Properties of a source. (Contributed by FL, 27-Jun-2014.)

Theorempropsrc 25280* Properties of a source. (Contributed by FL, 30-May-2014.)

Syntaxcsnk 25281 Extend class notation with the class of all sinks.

Definitiondf-sink 25282* A sink is a family of morphims indexed by a set which all have the same codomain. Joy of Cats, def. 10.62, p. 184. Experimental. (Contributed by FL, 15-Jul-2012.)

Syntaxcntrl 25283 Extend class notation with the class of all natural sources.

Definitiondf-natur 25284* A diagram is an indexed family of objects and morphisms in a category C. Maybe more than one morphism between two given objects, maybe none. One might choose a simple set with no structure for the set of indices as usual but we can also use a category. Let's call I this category. Morphisms of I are used as indices of morphisms of C and objects of I are used as indices of objects of C. With this convention a diagram is now a functor .

Using a category for the indices is even a better solution than using a simple set because with the functions dom_ and cod_ it is easy to express the relationship between a morphism in C and its attached objects simply by naming the relationship between the morphism in I and its attached objects (let's recall a functor preserves the domain and codomain).

So ...

Let be a diagram, a source indexed by the objects of is said to be natural for the diagram provided that each morphism of and the morphisms of the source connected to its domain and codomain commute. Joy of Cats, def. 11.3, (1) p. 193. Goldblatt calls "cone" what Adamek, Herrlich and Strecker call "natural source". Experimental. (Contributed by FL, 27-Jun-2014.)

Theoremisntr 25285* The predicate "is a natural source". (Contributed by FL, 27-Jun-2014.)

18.13.55  Limits and co-limits

Syntaxclmct 25286 Extend class notation with the class of all limits.

Definitiondf-limcat 25287* A limit of a diagram is a natural source for the diagram with the universal property that every natural source for uniquely factors through it. Joy of Cats, def. 11.3 (2), p. 194. Experimental. (Contributed by FL, 27-Jun-2014.)

Theoremislimcat 25288* The predicate "is a limit of a diagram." (Contributed by FL, 27-Jun-2014.)

18.13.56  Product and sum of two objects

Syntaxcprodo 25289 Extend class notation with the class of all object products.

Definitiondf-prodobj 25290* " A product in a category of two objects and is a -object together with a pair ( , ) of -arrows such that for any pair of -arrows of the form (, ) there is exactly one arrow such that and ". Goldblatt p. 47. Experimental. (Contributed by FL, 15-Jul-2012.)

Syntaxcsumo 25291 Extend class notation with the class of all object sums.

Definitiondf-sumobj 25292* " A co-product of C-objects and is a C-object together with a pair ( , ) of C-arrows such that for any pair of C-arrows of the form (, ) there is exactly one arrow such that and ". Goldblatt p. 54. Experimental. (Contributed by FL, 15-Jul-2012.)

18.13.57  Tarski's classes

Syntaxctar 25293 Extends class notation to include function .

Definitiondf-tar 25294* A function to study Tarski's classes. See valdom 25296 for its domain, vtare 25297 for its value at , vtarsu 25298 for its value at a successor, vtarl 25299 for its value at a limit ordinal. (Contributed by FL, 20-Mar-2011.)

Theoremvaltar 25295* The function as a recursive function. (Contributed by FL, 20-Mar-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremvaldom 25296 The domain of the function is the ordinal . (Contributed by FL, 20-Mar-2011.)

Theoremvtare 25297 Value of at . JFM CLASSES1 th. 10. (Contributed by FL, 20-Mar-2011.)

Theoremvtarsu 25298* The parts and the powersets of the elements of are elements of . As well as the parts of when they are elements of the smallest Tarski's class of which is an element. JFM CLASSES1 th. 11. (Contributed by FL, 20-Mar-2011.)

Theoremvtarl 25299 The value of at a limit ordinal. JFM CLASSES1 th. 12. (Contributed by FL, 20-Mar-2011.)

Theoremtartarmap 25300 The sequence has its values in a Tarski's class. (Contributed by FL, 20-Mar-2011.)

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