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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssiun3 24001* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)

Theoremssiun2sf 24002 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)

Theoremiuninc 24003* The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)

Theoremiundifdifd 24004* The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)

Theoremiundifdif 24005* The intersection of a set is the complement of the union of the complements. TODO shorten using iundifdifd 24004 (Contributed by Thierry Arnoux, 4-Sep-2016.)

Theoremiunrdx 24006* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)

19.3.3.7  Disjointness - misc additions

Theoremcbvdisjf 24007* Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Disj Disj

Theoremdisjss1f 24008 A subset of a disjoint collection is disjoint. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Disj Disj

Theoremdisjdifprg 24009* A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Disj

Theoremdisjdifprg2 24010* A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Disj

Theoremdisji2f 24011* Property of a disjoint collection: if and , and , then and are disjoint. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Disj

Theoremdisjif 24012* Property of a disjoint collection: if and have a common element , then . (Contributed by Thierry Arnoux, 30-Dec-2016.)
Disj

Theoremdisjorf 24013* Two ways to say that a collection for is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Disj

Theoremdisjorsf 24014* Two ways to say that a collection for is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)
Disj

Theoremdisjif2 24015* Property of a disjoint collection: if and have a common element , then . (Contributed by Thierry Arnoux, 6-Apr-2017.)
Disj

Theoremdisjabrex 24016* Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.)
Disj Disj

Theoremdisjabrexf 24017* Rewriting a disjoint collection into a partition of its image set (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
Disj Disj

Theoremdisjpreima 24018* A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
Disj Disj

Theoremdisjin 24019 If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 14-Feb-2017.)
Disj Disj

Theoremdisjxpin 24020* Derive a disjunction over a cross product from the disjunctions over its first and second elements. (Contributed by Thierry Arnoux, 9-Mar-2018.)
Disj        Disj        Disj

Theoremiundisjf 24021* Rewrite a countable union as a disjoint union. Cf. iundisj 19434 (Contributed by Thierry Arnoux, 31-Dec-2016.)
..^

Theoremiundisj2f 24022* A disjoint union is disjoint. Cf. iundisj2 19435 (Contributed by Thierry Arnoux, 30-Dec-2016.)
Disj ..^

Theoremdisjrdx 24023* Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)
Disj Disj

Theoremdisjex 24024* Two ways to say that two classes are disjoint (or equal). (Contributed by Thierry Arnoux, 4-Oct-2016.)

Theoremdisjexc 24025* A variant of disjex 24024, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)

19.3.4  Relations and Functions

19.3.4.1  Relations - misc additions

Theoremdfrel4 24026* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5764 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)

Theoremrnpropg 24027 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremxpdisjres 24028 Restriction of a constant function (or other cross product) outside of its domain (Contributed by Thierry Arnoux, 25-Jan-2017.)

Theoremcsbcnvg 24029 Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)

Theoremimadifxp 24030 Image of the difference with a cross product (Contributed by Thierry Arnoux, 13-Dec-2017.)

19.3.4.2  Functions - misc additions

Theoremfdmrn 24031 A different way to write is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremnvof1o 24032 An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremf1o3d 24033* Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)

Theoremrinvf1o 24034 Sufficient conditions for the restriction of an involution to be a bijection (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremdfimafnf 24035* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)

Theoremfunimass4f 24036 Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)

Theoremmptcnv 24037* The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)

Theoremfneq12 24038 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Theoremfimacnvinrn 24039 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)

Theoremfimacnvinrn2 24040 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)

Theoremsuppss2f 24041* Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.)

Theoremfovcld 24042 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)

Theoremelovimad 24043 Elementhood of the image set of an operation value (Contributed by Thierry Arnoux, 13-Mar-2017.)

Theoremofrn 24044 The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)

Theoremofrn2 24045 The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Theoremoff2 24046* The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)

Theoremofresid 24047 Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)

Theoremunipreima 24048* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)

Theoremsspreima 24049 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)

Theoremopfv 24050 Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremxppreima 24051 The preimage of a cross product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)

Theoremxppreima2 24052* The preimage of a cross product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)

Theoremfmptapd 24053* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfmptpr 24054* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremelunirn2 24055 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)

Theoremabfmpunirn 24056* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)

Theoremrabfmpunirn 24057* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)

Theoremabfmpeld 24058* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)

Theoremabfmpel 24059* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)

Theoremcbvmptf 24060* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)

TheoremfmptdF 24061 Domain and co-domain of the mapping operation; deduction form. This version of fmptd 5885 usex bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Theoremfmpt3d 24062* Domain and co-domain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Theoremresmptf 24063 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Theoremfvmpt2f 24064 Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)

Theoremmptfnf 24065 The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)

Theoremfnmptf 24066 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)

Theoremfeqmptdf 24067 Deduction form of dffn5f 5773. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)

Theoremfmptcof2 24068* Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)

Theoremfcomptf 24069* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 5896. (Contributed by Thierry Arnoux, 30-Jun-2017.)

Theoremcofmpt 24070* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)

Theoremofoprabco 24071* Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Theoremoffval2f 24072* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)

Theoremofpreima 24073* Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)

TheoremfuncnvmptOLD 24074* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremfuncnvmpt 24075* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)

Theoremfuncnv5mpt 24076* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)

Theoremfuncnv4mpt 24077* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Theoremrnmpt2ss 24078* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)

Theorempartfun 24079 Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.)

19.3.4.3  Isomorphisms - misc. add.

Theoremgtiso 24080 Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)

Theoremisoun 24081* Infer an isomorphism from for a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)

19.3.4.4  Disjointness (additional proof requiring functions)

Theoremdisjdsct 24082* A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 5503) (Contributed by Thierry Arnoux, 28-Feb-2017.)
Disj

19.3.4.5  First and second members of an ordered pair - misc additions

Theoremdf1stres 24083* Definition for a restriction of the (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theoremdf2ndres 24084* Definition for a restriction of the (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theorem1stnpr 24085 Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Theorem2ndnpr 24086 Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Theorem1stpreima 24087 The preimage by is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)

Theorem2ndpreima 24088 The preimage by is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)

Theoremcurry2ima 24089* The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)

19.3.4.6  Supremum - misc additions

Theoremsupssd 24090* Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)

19.3.4.7  Countable Sets

Theoremnnct 24091 is countable (Contributed by Thierry Arnoux, 29-Dec-2016.)

Theoremctex 24092 A countable set is a set (Contributed by Thierry Arnoux, 29-Dec-2016.)

Theoremssct 24093 The subset of a countable set is countable (Contributed by Thierry Arnoux, 31-Jan-2017.)

Theoremxpct 24094 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremsnct 24095 A singleton is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)

Theoremprct 24096 An unordered pair is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)

Theoremfnct 24097 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)

Theoremdmct 24098 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)

Theoremcnvct 24099 If a set is countable, its converse is as well. (Contributed by Thierry Arnoux, 29-Dec-2016.)

Theoremrnct 24100 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)

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