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Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisfunb 25001* The predicate "is a functor" . (Contributed by FL, 10-Feb-2008.)

Theoremfmamo 25002 A functor is a mapping between morphisms. (Contributed by FL, 10-Feb-2008.)

Theoremvidfunid 25003* The functor associates every object of to an object in . For the identification of objects with the identities see df-funcOLD 24999. JFM CAT1 th. 97. (Contributed by FL, 10-Feb-2008.)

Theoremiddvvidd 25004* Functors preserve domains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)

Theoremidcvvidc 25005* Functors preserve codomains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)

Theoremfmp 25006* Functors preserve morphisms composition. JFM CAT1 th. 99. (Contributed by FL, 2-Aug-2009.)

Theoremidfisf 25007 The identity functor is a functor. (Contributed by FL, 15-Jul-2008.)

Definitiondf-isof 25008* Class of isomorphisms. (Contributed by FL, 21-May-2012.)

16.11.52  Subcategories

Syntaxcsubcat 25009 Extend class notation with a function returning all the subcategories of a given category.

Definitiondf-subcat 25010 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 25011 The set of all the subcategories of . (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremissubcata 25012 The predicate "is a subcategory of" . (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremissubcatb 25013 The predicate "is a subcategory of" . (Contributed by FL, 17-Sep-2009.)

Theorembesubbeca 25014 Lemma to simplify some subcategories related theorems . (Contributed by FL, 17-Sep-2009.)

Theoremcatsbc 25015 A category belongs to the set of its subcategories. (Contributed by FL, 17-Sep-2009.)

Theoremobsubc2 25016 The objects of a subcategory are a subset of those of the supercategory. JFM CAT2 th. 11 . (Contributed by FL, 17-Sep-2009.)

Theoremidsubc 25017 The identity function of a subcategory is a subset of the identity function of the supercategory. (Contributed by FL, 17-Sep-2009.)

Theoremdomsubc 25018 The domain function of a subcategory is a subset of the domain function of the supercategory. (Contributed by FL, 19-Sep-2009.)

Theoremcodsubc 25019 The codomain function of a subcategory is a subset of the codomain function of the supercategory. (Contributed by FL, 19-Sep-2009.)

Theoremsubctct 25020 A subcategory is a category. (Contributed by FL, 17-Sep-2009.)

Theoremmorsubc 25021 The morphisms of a subcategory are a subset of those of the supercategory. (Contributed by FL, 18-Sep-2009.)

Theoremcmpsubc 25022 The composition law of a subcategory is a subset of the composition law of the supercategory. (Contributed by FL, 20-Sep-2009.)

Theoremidsubidsup 25023* 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 25024 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 25025 The inclusion functor is an embedding. (Contributed by FL, 2-Nov-2009.)

16.11.53  Terminal and initial objects

Syntaxciobj 25026 Extend class notation with the class of all initial objects.

Definitiondf-inob 25027* Definition of the initial objects of a category. Experimental. (Contributed by FL, 27-Jun-2014.)

Theoremisinob 25028* The predicate "are the initial objects of a category". (Contributed by FL, 27-Jun-2014.)

Syntaxctobj 25029 Extend class notation with the class of all terminal objects.

Definitiondf-termob 25030* Definition of the terminal objects of a category. Experimental. (Contributed by FL, 15-Jul-2012.)

16.11.54  Sources and sinks

Syntaxcsrce 25031 Extend class notation with the class of all sources.

Definitiondf-source 25032* 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 25033* Properties of a source. (Contributed by FL, 27-Jun-2014.)

Theorempropsrc 25034* Properties of a source. (Contributed by FL, 30-May-2014.)

Syntaxcsnk 25035 Extend class notation with the class of all sinks.

Definitiondf-sink 25036* 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 25037 Extend class notation with the class of all natural sources.

Definitiondf-natur 25038* 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 25039* The predicate "is a natural source". (Contributed by FL, 27-Jun-2014.)

16.11.55  Limits and co-limits

Syntaxclmct 25040 Extend class notation with the class of all limits.

Definitiondf-limcat 25041* 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 25042* The predicate "is a limit of a diagram." (Contributed by FL, 27-Jun-2014.)

16.11.56  Product and sum of two objects

Syntaxcprodo 25043 Extend class notation with the class of all object products.

Definitiondf-prodobj 25044* " 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 25045 Extend class notation with the class of all object sums.

Definitiondf-sumobj 25046* " 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.)

16.11.57  Tarski's classes

Syntaxctar 25047 Extends class notation to include function .

Definitiondf-tar 25048* A function to study Tarski's classes. See valdom 25050 for its domain, vtare 25051 for its value at , vtarsu 25052 for its value at a successor, vtarl 25053 for its value at a limit ordinal. (Contributed by FL, 20-Mar-2011.)

Theoremvaltar 25049* The function as a recursive function. (Contributed by FL, 20-Mar-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremvaldom 25050 The domain of the function is the ordinal . (Contributed by FL, 20-Mar-2011.)

Theoremvtare 25051 Value of at . JFM CLASSES1 th. 10. (Contributed by FL, 20-Mar-2011.)

Theoremvtarsu 25052* 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 25053 The value of at a limit ordinal. JFM CLASSES1 th. 12. (Contributed by FL, 20-Mar-2011.)

Theoremtartarmap 25054 The sequence has its values in a Tarski's class. (Contributed by FL, 20-Mar-2011.)

Theorempwtsm 25055 If belongs to the smallest Tarski's class that contains so does . CLASSES1 th. 7. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Theoremsubtsm 25056 If belongs to the smallest Tarski's class that contains so does the subsets of . CLASSES1. th. 6. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Theoremsubtareqbe 25057 If is a subset of the smallest Tarski's class that contains then it is equipotent to this class or it belongs to it. CLASSES1 th. 8. (Contributed by FL, 17-Apr-2011.)

Syntaxctr 25058 Extend class notation to include the function whose value is the transitive closure of its operand.

Definitiondf-trcls 25059* The transitive closure of a set. (Contributed by FL, 17-Apr-2011.)

Theoremtrclval 25060* The transitive closure of a set A. (Contributed by FL, 17-Apr-2011.)

Theoremvtarsuelt 25061* C belongs to the value of at a successor of iff it is a part of at , the powerset of an element or a part of an element of at . CLASSES1 th. 13 (Contributed by FL, 13-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Theorempartarelt1 25062 If is a part of an element of our tar function at then is an element or tar at . CLASSES1 th. 14 (Contributed by FL, 13-Apr-2011.)

Theorempartarelt2 25063 If is an element of our tar function at then is an element or tar at . CLASSES1 th. 15 (Contributed by FL, 13-Apr-2011.)

Theoremtareltsuc 25064 All the element of at are elements of at . CLASSES1 th. 18 (Contributed by FL, 13-Apr-2011.)

Theoremeltintpar 25065 An element of the intersection of a Tarski's class with the class of the ordinal numbers is a part of the intersection. (Contributed by FL, 20-Apr-2011.)

Theoreminttaror 25066 The intersection of a Tarski's class with the class of the ordinal numbers is an ordinal number. (Contributed by FL, 20-Apr-2011.)

Theoreminttarcar 25067 The intersection of a Tarski's class and the ordinal numbers is equipotent to the Tarski's class. JFM CLASSES2. . (Contributed by FL, 20-Apr-2011.)

Theoremcarinttar 25068 The cardinal of the intersection of a Tarski's class with the class of the ordinal numbers. (Contributed by FL, 20-Apr-2011.)

Theoremcarinttar2 25069 The cardinal of a Tarski's class equals the intersection of the Tarski's class with the class of the ordinal numbers. CLASSES2 th. 10. (Contributed by FL, 20-Apr-2011.)

Theoremcardtar 25070 The cardinal of an element of a Tarski's class belongs to the Tarski's class. th. 12 CLASSES2 (Contributed by FL, 20-Apr-2011.)

Theoremcartarlim 25071 The cardinal of a Tarski's class is a limit ordinal. CLASSES2 th. 21. (Contributed by FL, 20-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Theoremelcarelcl 25072 An element of the cardinal of the Tarski's class is an element of . th. 14 CLASSES2. (Contributed by FL, 20-Nov-2011.)

Theoremfnctartar 25073 Consider functions whose domain is an element of a transitive Tarski's class and whose range is , then they are elements of . CLASSES2 th. 23. (Contributed by FL, 26-Sep-2011.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremfnctartar2 25074 Consider functions whose domain is an element and a part of a Tarski's class and whose range is , then they are elements of . CLASSES2 th. 23. (Contributed by FL, 27-Sep-2011.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremfnctartar3 25075 If the cardinal of of a part of is less than . a function from to is a part of . CLASSES2 th. 23. (Contributed by FL, 20-Nov-2011.)

16.11.58  Category Set

Syntaxccmrcase 25076 Extend class notation to include the morphisms of the category Set.

Definitiondf-morcatset 25077* The morphisms of the category Set. ( is redundant and could be retrieved from .) Experimental. (Contributed by FL, 15-Sep-2013.)

Theoremprismorcsetlem 25078* Lemma for prismorcset 25080. (Contributed by FL, 15-Sep-2013.)

Theoremprismorcsetlemb 25079* Lemma for prismorcset 25080. First use of the property of a universe through grumap 8310. (Contributed by FL, 6-Nov-2013.)

Theoremprismorcset 25080 The predicate "is a morphism of the category Set". (Contributed by FL, 15-Sep-2013.)

Theoremmorcatset1 25081* The morphisms of the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremdfiunv2 25082* Define double indexed union. (Contributed by FL, 6-Nov-2013.)

Theoremprismorcsetlemc 25083* Lemma for morexcmp 25133. (Contributed by FL, 6-Nov-2013.)

Theoremprismorcset2 25084 The predicate "is a morphism of the category Set". (Contributed by FL, 15-Sep-2013.)

Syntaxcdomcase 25085 Extend class notation to include the domain of a morphism in the category Set.

Definitiondf-domcatset 25086* The domain of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)

Syntaxcgraphcase 25087 Extend class notation to include the graph of a morphism in the category Set.

Definitiondf-graphcatset 25088* The underlying function of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)

Theoremisgraphmrph 25089 The graph of a morhism in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremisgraphmrph2 25090 The graph of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
.graph        .Morphism        .Morphism .graph

Theoremdomcatfun 25091 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremdomdomcatfun 25092 The domain of the function in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremdomdomcatfun1 25093 The domain of the function in the category Set. (Contributed by FL, 6-Nov-2013.)
.dom        .Morphism        .dom .Morphism

Theoremdomcatsetval 25094 The domain of a morphism in the category Set is a member of the underlying universe. (Contributed by FL, 6-Nov-2013.)

Theoremdomcatsetval2 25095 The domain of a morphism in the category Set is a member of the underlying universe. (Contributed by FL, 6-Nov-2013.)
.Morphism        .dom        .Morphism .dom

Theoremdomcatval 25096 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremdomcatval2 25097 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
.Morphism        .dom        .Morphism .dom

Syntaxccodcase 25098 Extend class notation to include the codomain of a morphism in the category Set.

Definitiondf-codcatset 25099* The codomain of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)

Theoremcodcatfun 25100 The codomain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)

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