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Type | Label | Description |
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Statement | ||
Theorem | resiima 5201 | The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.) |
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Theorem | ima0 5202 | Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.) |
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Theorem | 0ima 5203 | Image under the empty relation. (Contributed by FL, 11-Jan-2007.) |
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Theorem | csbima12 5204 | Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.) |
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Theorem | csbima12gOLD 5205 | Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) Obsolete as of 20-Aug-2018. Use csbfv12 5923 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | imadisj 5206 | A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.) |
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Theorem | cnvimass 5207 | A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.) |
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Theorem | cnvimarndm 5208 | The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.) |
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Theorem | imasng 5209* | The image of a singleton. (Contributed by NM, 8-May-2005.) |
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Theorem | relimasn 5210* | The image of a singleton. (Contributed by NM, 20-May-1998.) |
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Theorem | elrelimasn 5211 | Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.) |
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Theorem | elimasn 5212 | Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | elimasng 5213 | Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) |
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Theorem | elimasni 5214 | Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.) |
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Theorem | args 5215* |
Two ways to express the class of unique-valued arguments of ![]() ![]() ![]() ![]() |
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Theorem | eliniseg 5216 |
Membership in an initial segment. The idiom ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | epini 5217 | Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.) |
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Theorem | iniseg 5218* | An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) |
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Theorem | inisegn0 5219 | Nonemptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
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Theorem | dffr3 5220* | Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.) |
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Theorem | dfse2 5221* | Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.) |
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Theorem | imass1 5222 | Subset theorem for image. (Contributed by NM, 16-Mar-2004.) |
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Theorem | imass2 5223 | Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.) |
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Theorem | ndmima 5224 | The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) (Proof shortened by OpenAI, 3-Jul-2020.) |
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Theorem | ndmimaOLD 5225 | The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) Obsolete version of ndmima 5224 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | relcnv 5226 | A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.) |
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Theorem | relbrcnvg 5227 |
When ![]() |
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Theorem | eliniseg2 5228 | Eliminate the class existence constraint in eliniseg 5216. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.) |
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Theorem | relbrcnv 5229 |
When ![]() |
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Theorem | cotrg 5230* | Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5231 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5231. (Revised by Richard Penner, 24-Dec-2019.) |
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Theorem | cotr 5231* | Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 5230. (Contributed by NM, 27-Dec-1996.) |
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Theorem | issref 5232* | Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.) |
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Theorem | cnvsym 5233* | Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | intasym 5234* | Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | asymref 5235* |
Two ways of saying a relation is antisymmetric and reflexive.
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Theorem | asymref2 5236* | Two ways of saying a relation is antisymmetric and reflexive. (Contributed by NM, 6-May-2008.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) |
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Theorem | intirr 5237* | Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | brcodir 5238* | Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.) |
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Theorem | codir 5239* | Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.) |
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Theorem | qfto 5240* | A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.) |
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Theorem | xpidtr 5241 |
A square Cartesian product ![]() ![]() ![]() ![]() ![]() |
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Theorem | trin2 5242 | The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.) |
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Theorem | poirr2 5243 | A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.) |
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Theorem | trinxp 5244 | The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square Cartesian product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.) |
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Theorem | soirri 5245 | A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
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Theorem | sotri 5246 | A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
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Theorem | son2lpi 5247 | A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
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Theorem | sotri2 5248 |
A transitivity relation. (Read ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sotri3 5249 |
A transitivity relation. (Read ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | poleloe 5250 | Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | poltletr 5251 | Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | somin1 5252 | Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | somincom 5253 | Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | somin2 5254 | Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | soltmin 5255 | Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | cnvopab 5256* | The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | mptcnv 5257* | The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.) |
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Theorem | cnv0 5258 | The converse of the empty set. (Contributed by NM, 6-Apr-1998.) |
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Theorem | cnvi 5259 | The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | cnvun 5260 | The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | cnvdif 5261 | Distributive law for converse over class difference. (Contributed by Mario Carneiro, 26-Jun-2014.) |
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Theorem | cnvin 5262 | Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.) |
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Theorem | rnun 5263 | Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.) |
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Theorem | rnin 5264 | The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) |
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Theorem | rniun 5265 | The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.) |
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Theorem | rnuni 5266* | The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.) |
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Theorem | imaundi 5267 | Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.) |
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Theorem | imaundir 5268 | The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.) |
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Theorem | dminss 5269 | An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.) |
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Theorem | imainss 5270 | An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.) |
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Theorem | inimass 5271 | The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
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Theorem | inimasn 5272 | The intersection of the image of singleton. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
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Theorem | cnvxp 5273 | The converse of a Cartesian product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | xp0 5274 | The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
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Theorem | xpnz 5275 | The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.) |
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Theorem | xpeq0 5276 | At least one member of an empty Cartesian product is empty. (Contributed by NM, 27-Aug-2006.) |
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Theorem | xpdisj1 5277 | Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
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Theorem | xpdisj2 5278 | Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
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Theorem | xpsndisj 5279 | Cartesian products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.) |
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Theorem | difxp 5280 | Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.) |
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Theorem | difxp1 5281 | Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.) |
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Theorem | difxp2 5282 | Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.) |
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Theorem | djudisj 5283* | Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
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Theorem | xpdifid 5284* | The set of distinct couples in a Cartesian product. (Contributed by Thierry Arnoux, 25-May-2019.) |
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Theorem | resdisj 5285 | A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | rnxp 5286 | The range of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
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Theorem | dmxpss 5287 | The domain of a Cartesian product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.) |
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Theorem | rnxpss 5288 | The range of a Cartesian product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | rnxpid 5289 | The range of a square Cartesian product. (Contributed by FL, 17-May-2010.) |
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Theorem | ssxpb 5290 | A Cartesian product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.) |
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Theorem | xp11 5291 | The Cartesian product of nonempty classes is one-to-one. (Contributed by NM, 31-May-2008.) |
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Theorem | xpcan 5292 | Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.) |
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Theorem | xpcan2 5293 | Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.) |
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Theorem | ssrnres 5294 | Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.) |
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Theorem | rninxp 5295* | Range of the intersection with a Cartesian product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | dminxp 5296* | Domain of the intersection with a Cartesian product. (Contributed by NM, 17-Jan-2006.) |
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Theorem | imainrect 5297 | Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.) |
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Theorem | xpima 5298 | The image by a constant function (or other Cartesian product). (Contributed by Thierry Arnoux, 4-Feb-2017.) |
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Theorem | xpima1 5299 | The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
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Theorem | xpima2 5300 | The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
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