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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremimaeq2 5201 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq1i 5202 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq2i 5203 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq1d 5204 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq2d 5205 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq12d 5206 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)

Theoremdfima2 5207* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdfima3 5208* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelimag 5209* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)

Theoremelima 5210* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)

Theoremelima2 5211* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)

Theoremelima3 5212* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)

Theoremnfima 5213 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremnfimad 5214 Deduction version of bound-variable hypothesis builder nfima 5213. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbima12g 5215 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.)

Theoremcsbima12gALT 5216 Move class substitution in and out of the image of a function. (This is csbima12g 5215 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbima12gALTVD 29071. Although the proof is shorter, the total number of steps of all theorems used in the proof is probably longer. (Contributed by NM, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremimadmrn 5217 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)

Theoremimassrn 5218 The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)

Theoremimaexg 5219 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)

Theoremimai 5220 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Theoremrnresi 5221 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresiima 5222 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

Theoremima0 5223 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)

Theorem0ima 5224 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)

Theoremimadisj 5225 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)

Theoremcnvimass 5226 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)

Theoremcnvimarndm 5227 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremimasng 5228* The image of a singleton. (Contributed by NM, 8-May-2005.)

Theoremrelimasn 5229* The image of a singleton. (Contributed by NM, 20-May-1998.)

Theoremelrelimasn 5230 Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)

Theoremelimasn 5231 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelimasng 5232 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)

Theoremelimasni 5233 Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)

Theoremargs 5234* Two ways to express the class of unique-valued arguments of , which is the same as the domain of whenever is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg " for this class (for which we have no separate notation). Observe the resemblance to the alternative definition dffv4 5727 of function value, which is based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)

Theoremeliniseg 5235 Membership in an initial segment. The idiom , meaning , is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremepini 5236 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoreminiseg 5237* 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.)

Theoremdffr3 5238* 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.)

Theoremdfse2 5239* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se

Theoremexse2 5240 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se

Theoremimass1 5241 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Theoremimass2 5242 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)

Theoremndmima 5243 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)

Theoremrelcnv 5244 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)

Theoremrelbrcnvg 5245 When is a relation, the sethood assumptions on brcnv 5057 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremeliniseg2 5246 Eliminate the class existence constraint in eliniseg 5235. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)

Theoremrelbrcnv 5247 When is a relation, the sethood assumptions on brcnv 5057 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremcotr 5248* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremissref 5249* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)

Theoremcnvsym 5250* 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.)

Theoremintasym 5251* 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.)

Theoremasymref 5252* Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 5397. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremasymref2 5253* Two ways of saying a relation is antisymmetric and reflexive. (Contributed by NM, 6-May-2008.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Theoremintirr 5254* 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.)

Theorembrcodir 5255* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)

Theoremcodir 5256* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremqfto 5257* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremxpidtr 5258 A square cross product is a transitive relation. (Contributed by FL, 31-Jul-2009.)

Theoremtrin2 5259 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)

Theorempoirr2 5260 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)

Theoremtrinxp 5261 The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square cross product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)

Theoremsoirri 5262 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri 5263 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremson2lpi 5264 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri2 5265 A transitivity relation. (Read and implies .) (Contributed by Mario Carneiro, 10-May-2013.)

Theoremsotri3 5266 A transitivity relation. (Read and implies .) (Contributed by Mario Carneiro, 10-May-2013.)

TheoremsoirriOLD 5267 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremsotriOLD 5268 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremson2lpiOLD 5269 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorempoleloe 5270 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theorempoltletr 5271 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremsomin1 5272 Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremsomincom 5273 Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremsomin2 5274 Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremsoltmin 5275 Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremcnvopab 5276* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnv0 5277 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)

Theoremcnvi 5278 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.)

Theoremcnvun 5279 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.)

Theoremcnvdif 5280 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremcnvin 5281 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.)

Theoremrnun 5282 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Theoremrnin 5283 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremrniun 5284 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremrnuni 5285* 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.)

Theoremimaundi 5286 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremimaundir 5287 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 5288 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.)

Theoremimainss 5289 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Theoreminimass 5290 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoreminimasn 5291 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremcnvxp 5292 The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremxp0 5293 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)

Theoremxpnz 5294 The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)

Theoremxpeq0 5295 At least one member of an empty cross product is empty. (Contributed by NM, 27-Aug-2006.)

Theoremxpdisj1 5296 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpdisj2 5297 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpsndisj 5298 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)

Theoremdjudisj 5299* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremresdisj 5300 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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