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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminttarcar 25901 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 25902 The cardinal of the intersection of a Tarski's class with the class of the ordinal numbers. (Contributed by FL, 20-Apr-2011.)

Theoremcarinttar2 25903 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 25904 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 25905 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 25906 An element of the cardinal of the Tarski's class is an element of . th. 14 CLASSES2. (Contributed by FL, 20-Nov-2011.)

Theoremfnctartar 25907 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 25908 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 25909 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.)

18.13.58  Category Set

Syntaxccmrcase 25910 Extend class notation to include the morphisms of the category Set.

Definitiondf-morcatset 25911* The morphisms of the category Set. ( is redundant and could be retrieved from .) Experimental. (Contributed by FL, 15-Sep-2013.)

Theoremprismorcsetlem 25912* Lemma for prismorcset 25914. (Contributed by FL, 15-Sep-2013.)

Theoremprismorcsetlemb 25913* Lemma for prismorcset 25914. First use of the property of a universe through grumap 8430. (Contributed by FL, 6-Nov-2013.)

Theoremprismorcset 25914 The predicate "is a morphism of the category Set". (Contributed by FL, 15-Sep-2013.)

Theoremmorcatset1 25915* The morphisms of the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremdfiunv2 25916* Define double indexed union. (Contributed by FL, 6-Nov-2013.)

Theoremprismorcsetlemc 25917* Lemma for morexcmp 25967. (Contributed by FL, 6-Nov-2013.)

Theoremprismorcset2 25918 The predicate "is a morphism of the category Set". (Contributed by FL, 15-Sep-2013.)

Syntaxcdomcase 25919 Extend class notation to include the domain of a morphism in the category Set.

Definitiondf-domcatset 25920* The domain of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)

Syntaxcgraphcase 25921 Extend class notation to include the graph of a morphism in the category Set.

Definitiondf-graphcatset 25922* The underlying function of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)

Theoremisgraphmrph 25923 The graph of a morhism in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremisgraphmrph2 25924 The graph of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
.graph        .Morphism        .Morphism .graph

Theoremdomcatfun 25925 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremdomdomcatfun 25926 The domain of the function in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremdomdomcatfun1 25927 The domain of the function in the category Set. (Contributed by FL, 6-Nov-2013.)
.dom        .Morphism        .dom .Morphism

Theoremdomcatsetval 25928 The domain of a morphism in the category Set is a member of the underlying universe. (Contributed by FL, 6-Nov-2013.)

Theoremdomcatsetval2 25929 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 25930 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremdomcatval2 25931 The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
.Morphism        .dom        .Morphism .dom

Syntaxccodcase 25932 Extend class notation to include the codomain of a morphism in the category Set.

Definitiondf-codcatset 25933* The codomain of a morphism in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)

Theoremcodcatfun 25934 The codomain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremcodcatsetval 25935 The codomain of a morphism in the category Set is a member of the underlying universe. (Contributed by FL, 6-Nov-2013.)

Theoremcodcatval 25936 The codomain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremcodcatval2 25937 The codomain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
.Morphism        .cod        .Morphism .cod

Theoremprismorcset3 25938 The predicate "is a morphism of the category Set". (Contributed by FL, 6-Nov-2013.)
.dom        .cod        .graph        .Morphism        .Morphism .graph .cod .dom

Syntaxcidcase 25939 Extend class notation to include the identity morphism of an object in the category Set.

Definitiondf-idcatset 25940* The identity morphims in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)

Theoremidcatfun 25941 The identity morphims in the category Set. (Contributed by FL, 6-Nov-2013.)

Theoremobcatset 25942 The objects of the category Set is the Universe. (Contributed by FL, 6-Nov-2013.)
.Object        .Object

Theoremidcatval 25943 An identity morphism is a morphism. (Contributed by FL, 6-Nov-2013.)

Theoremidcatval2 25944 An identity morphism is a morphism. (Contributed by FL, 6-Nov-2013.)
.id        .Morphism        .id .Morphism

Theoremdomidcat 25945 The underlying universe of an identity morphism. (Contributed by FL, 6-Nov-2013.)

Theoremidmor 25946 An identity morphism. (Contributed by FL, 6-Nov-2013.)

Theoremidmorimor 25947 An identity morphism is a morphism. (Contributed by FL, 6-Nov-2013.)

Theoremdomidmor 25948 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)

Theoremdomidmor2 25949 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
.dom        .id        .dom .id

Theoremcodidmor 25950 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)

Theoremcodidmor2 25951 Domain of an identity morphism. (Contributed by FL, 6-Nov-2013.)
.cod        .id        .cod .id

Theoremgrphidmor 25952 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)

Theoremgrphidmor2 25953 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
.graph        .id        .graph .id

Theoremgrphidmor3 25954 Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
.graph        .id        .graph .id

Syntaxcrocase 25955 Extend class notation to include the morphisms composition in the category Set.

Definitiondf-rocatset 25956* Composition of two morphisms in the category Set. Experimental. (Contributed by FL, 6-Nov-2013.)

Theoremisrocatset 25957* Definition of the composition of two morphisms in the category Set . (Contributed by FL, 6-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremcmp2morp 25958 Composite of two morphisms. (Contributed by FL, 6-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremrocatval 25959 The composite of two morphisms in the category Set is a morphism. (Contributed by FL, 6-Nov-2013.)

Theoremrocatval2 25960 The composite of two morphisms in the category Set is a morphism. (Contributed by FL, 7-Nov-2013.)
.Morphism        .dom        .cod        .Morphism .Morphism .dom .cod .Morphism

Theoremcmp2morpcats 25961 Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
.Morphism        .dom        .cod        .Morphism .Morphism .dom .cod .dom .cod

Theoremcmp2morpcatt 25962 Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
.Morphism        .dom        .cod        .graph        .Morphism .Morphism .dom .cod .dom .cod .graph .graph

Theoremcmp2morpgrp 25963 Graph of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
.Morphism        .dom        .cod        .graph        .Morphism .Morphism .dom .cod .graph .graph .graph

Theoremcmp2morpdom 25964 Domain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
.Morphism        .dom        .cod        .Morphism .Morphism .dom .cod .dom .dom

Theoremcmp2morpcod 25965 Codomain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
.Morphism        .dom        .cod        .Morphism .Morphism .dom .cod .cod .cod

Theoremcmpmorass 25966 Associativity of composition in category Set. (Contributed by FL, 7-Nov-2013.)
.Morphism        .dom        .cod        .Morphism .Morphism .Morphism .dom .cod .dom .cod

Theoremmorexcmp 25967 A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
.Morphism        .dom        .cod        .graph        .Morphism .dom .cod .graph

Theoremmorexcmp2 25968 A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.)
.Morphism        .dom        .cod        .graph        .Morphism .dom .cod .graph

Theoremcmpidmor2 25969 Composition with an identity. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
.Morphism        .cod        .id        .Morphism .id .cod

Theoremcmpidmor3 25970 Composition with an identity. (Contributed by FL, 8-Nov-2013.)
.Morphism        .dom        .id        .Morphism .id .dom

Theoremcmpmorfun 25971 Composition of morphisms is a function. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremcmppar2 25972* Morphisms composition is defined every time the codomain of the second operand matches the domain of the first one. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremcmppar 25973 Composition of morphisms is a partial operation in the set of morphisms. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)

Theoremcmppar3 25974 Morphisms composition is defined every time the codomain of the second operand matches the domain of the first one. (Contributed by FL, 8-Nov-2013.)

Theoremcmpmor 25975 The composite of two morphisms is a morphism. (Contributed by FL, 8-Nov-2013.)

Syntaxccaset 25976 Extend class notation to include the category Set.

Definitiondf-catset 25977 Definition of the category Set. (We should say "the categories Set" since there is such a category per universe but for our purpose they are equivalent obviously.) Experimental. (Contributed by FL, 8-Nov-2013.)

Theoremiscatset 25978 The category Set. (Contributed by FL, 8-Nov-2013.)

Theoremsetiscat 25979 The category set is a category. (Contributed by FL, 6-Nov-2013.)

18.13.59  Grammars, Logics, Machines and Automata

Syntaxckln 25980 Extend class notation with the Kleene star.

18.13.60  Words

Syntaxcwrd 25981 Extend class notation with the class of words of a given size.

Definitiondf-words 25982* The words of size over an alphabet are the finite sequences over of size . Their domains are fiercely set to so that I can concatenate them easily. The case is a bit tricky and corresponds to the unique empty word (often denoted by an epsilon or by in textbooks.) Experimental. (Contributed by FL, 14-Jan-2014.)

Theoremisword 25983 The words over a set . (Contributed by FL, 14-Jan-2014.)

Syntaxcdwords 25984 Syntax for the dWords operator.
dWords

Definitiondf-dwords 25985* Words of size S over an alphabet with all the elements different. (For my private use only. Don't use.) (Contributed by FL, 26-May-2016.)
dWords

Theoremisnword 25986* The words over a set . (For my private use only. Don't use.) (Contributed by FL, 26-May-2014.)
dWords

Definitiondf-kle 25987* The Kleene star of an alphabet is the set of all the finite sequences of elements of this alphabet. Experimental. (Contributed by FL, 14-Jan-2014.)

TheoremisKleene 25988* The predicate is the Kleene star of a set . An element of is called a word. (Contributed by FL, 14-Jan-2014.)

Theorem1iskle 25989 Symbols and variables belong to the Kleene star of . (Contributed by FL, 2-Feb-2014.)

Theoremselsubf 25990 A way of selecting a subset of functions so that their values belong to . (Contributed by FL, 14-Jan-2014.)

Theoremselsubf3 25991 A way of selecting a subset of functions so that their values belong to . (Contributed by FL, 14-Jan-2014.)

Theoremselsubf3g 25992 A way of selecting a subset of functions so that their values belong to . (Contributed by FL, 14-Jan-2014.)

Syntaxclincl 25993 Extend class notation with the class of inductive closures.

Definitiondf-indcls 25994* Definition of an inductive closure. Top down definition. Gallier p. 19 (Contributed by FL, 14-Jan-2014.)

Theoremlemindclsbu 25995* Lemma for indcls2 25996. (Contributed by FL, 14-Jan-2014.)

Theoremindcls2 25996* The inductive closure of under . (Contributed by FL, 14-Jan-2014.)

Theoremxindcls 25997* X is a part of the inductive closure of under . (Contributed by FL, 15-Jan-2014.)

Syntaxcgrm 25998 Extend class notation with the class of all grammars.

Definitiondf-grm 25999* A grammar is a structure composed of a set of non-terminal symbols , of terminal symbols , a set of productions and a distinguished element of , the start symbol . Experimental. (Contributed by FL, 15-Jul-2012.)

Syntaxcsym 26000 Extend class notation with a function returning the symbols of a grammar.

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