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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsuppss2f 23201* Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.)

Theoremxpima 23202 The image of a constant function (or other cross product). (Contributed by Thierry Arnoux, 4-Feb-2017.)

Theoremfovcld 23203 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)

Theoremdfrel4 23204* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5568 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)

Theoremelovimad 23205 Elementhood of the image set of an operation value (Contributed by Thierry Arnoux, 13-Mar-2017.)

Theoremofrn 23206 The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)

Theoremofrn2 23207 The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Theoremoff2 23208* The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)

Theoremunipreima 23209* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)

Theoremsspreima 23210 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)

Theoremxppreima 23211 The preimage of a cross product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)

Theoremxppreima2 23212* The preimage of a cross product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)

Theoremfmptapd 23213* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfmptpr 23214* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremelunirn2 23215 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)

Theoremabfmpunirn 23216* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)

Theoremrabfmpunirn 23217* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)

Theoremabfmpeld 23218* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)

Theoremabfmpel 23219* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)

Theoremcbvmptf 23220* 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 23221 Domain and co-domain of the mapping operation; deduction form. This version of fmptd 5684 usex bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Theoremfmpt3d 23222* Domain and co-domain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Theoremresmptf 23223 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Theoremfvmpt2f 23224 Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)

Theoremfvmpt2d 23225* Deduction version of fvmpt2 5608. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Theoremmptfnf 23226 The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)

Theoremfnmptf 23227 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)

Theoremfeqmptdf 23228 Deduction form of dffn5f 5577. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)

Theoremfmptcof2 23229* 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 23230* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 5694. (Contributed by Thierry Arnoux, 30-Jun-2017.)

Theoremcofmpt 23231* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)

Theoremofoprabco 23232* Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Theoremoffval2f 23233* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)

TheoremfuncnvmptOLD 23234* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)

Theoremfuncnvmpt 23235* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)

Theoremfuncnv5mpt 23236* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)

Theoremfuncnv4mpt 23237* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Theoremrnmptss 23238* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremrnmpt2ss 23239* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)

Theorempartfun 23240 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.)

Theoremgtiso 23241 Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)

Theoremisoun 23242* Infer an isomorphism from for a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)

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

Theoremdf1stres 23243* Definition for a restriction of the (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theoremdf2ndres 23244* Definition for a restriction of the (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theorem1stnpr 23245 Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Theorem2ndnpr 23246 Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Theoremcurry2ima 23247* The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theoremsupssd 23248* Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)

18.3.14  Ordering on reals - misc additions

Theoremlt2addrd 23249* If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)

18.3.15  Extended reals - misc additions

Theoremxrlelttric 23250 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)

Theoremxrre3FL 23251 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.) (TODO remove and use xrre3 10500, which was imported )

Theoremxraddge02 23252 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)

Theoremxlt2addrd 23253* If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)

Theoremxrsupssd 23254 Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Theoremxrofsup 23255 The supremum is preserved by extended addition set operation. (provided minus infinity is not involved as it does not behave well with addition) (Contributed by Thierry Arnoux, 20-Mar-2017.)

Theoremsupxrnemnf 23256 The supremum of a non empty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Theoremxrhaus 23257 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
ordTop

18.3.16  Real number intervals - misc additions

Theoremicossicc 23258 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)

Theoremiocssicc 23259 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)

Theoremioossico 23260 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)

Theoremiocssioo 23261 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)

Theoremicossioo 23262 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)

Theoremicossico 23263 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremioossioo 23264 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)

Theoremjoiniooico 23265 Disjoint joining an open interval with a closed below, open above interval to form a closed below, open above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)

Theoremiccgelb 23266 An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)

Theoremsnunioc 23267 The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Theoremubico 23268 A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.)

Theoremxeqlelt 23269 Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.)

Theoremeliccelico 23270 Relate elementhood to a closed interval with elementhood to the same closed-below, opened-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)

Theoremelicoelioo 23271 Relate elementhood to a closed-below, opened-above interval with elementhood to the same opened interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)

Theoremiocinioc2 23272 Intersection between two opened below, closed above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.)

Theoremxrdifh 23273 Set difference of a half-opened interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)

Theoremiocinif 23274 Relate intersection of two opened below, closed above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)

Theoremdifioo 23275 The difference between two opened intervals sharing the same lower bound (Contributed by Thierry Arnoux, 26-Sep-2017.)

18.3.17  Finite intervals of integers - misc additions

Theoremfzssnn 23276 Finite sets of sequential integers starting from a natural are a subset of the natural numbers. (Contributed by Thierry Arnoux, 4-Aug-2017.)

Theoremssnnssfz 23277* For any finite subset of , find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.)

18.3.18  Half-open integer ranges - misc additions

Theoremfzossnn 23278 Half-opened integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
..^

Theoremelfzo1 23279 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
..^

18.3.19  Closed unit

Theoremunitsscn 23280 The closed unit is a subset of the set of the complex numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremelunitrn 23281 The closed unit is a subset of the set of the real numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)

Theoremelunitcn 23282 The closed unit is a subset of the set of the complext numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)

Theoremelunitge0 23283 An element of the closed unit is positive Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 20-Dec-2016.)

Theoremunitssxrge0 23284 The closed unit is a subset of the set of the extended non-negative reals. Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremunitdivcld 23285 Necessary conditions for a quotient to be in the closed unit. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)

Theoremiistmd 23286 The closed unit monoid is a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.)
mulGrpflds        TopMnd

18.3.20  Topology of ` ( RR X. RR ) `

Theoremtpr2tp 23287 The usual topology on is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)
TopOn

Theoremtpr2uni 23288 The usual topology on is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremclduni 23289 For any topology, the union of the closed sets is the base set. (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremxpinpreima 23290 Rewrite the cartesian product of two sets as the intersection of their preimage by and , the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Theoremxpinpreima2 23291 Rewrite the cartesian product of two sets as the intersection of their preimage by and , the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)

Theoremsqsscirc1 23292 The complex square of side is a subset of the complex circle of radius . (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theoremsqsscirc2 23293 The complex square of side is a subset of the complex circle of radius . (Contributed by Thierry Arnoux, 25-Sep-2017.)

Theoremcnre2csqlem 23294* Lemma for cnre2csqima 23295 (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theoremcnre2csqima 23295* Image of a centered square by the canonical bijection from to . (Contributed by Thierry Arnoux, 27-Sep-2017.)

Theoremtpr2rico 23296* For any point of an open set of the usual topology on there is a closed below opened above square which contains that point and is entirely in the open set. This is square is actually similar to a ball by the norm, closed below, centered on . (Contributed by Thierry Arnoux, 21-Sep-2017.)

18.3.21  Order topology - misc. additions

Theoremcnvordtrestixx 23297* The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.)
ordTop t ordTop

18.3.22  Continuity in topological spaces - misc. additions

Theoremressplusf 23298 The group operation function of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
s

Theoremmndpluscn 23299* A mapping that is both an homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.)
TopOn       TopOn

Theoremmhmhmeotmd 23300 Deduce a Topological Monoid using mapping that is both an homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
MndHom               TopMnd              TopMnd

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