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Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremimai 5001 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)

Theoremrnresi 5002 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresiima 5003 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)

Theoremima0 5004 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)

Theorem0ima 5005 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)

Theoremimadisj 5006 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)

Theoremcnvimass 5007 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)

Theoremcnvimarndm 5008 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremimasng 5009* The image of a singleton. (Contributed by NM, 8-May-2005.)

Theoremrelimasn 5010* The image of a singleton. (Contributed by NM, 20-May-1998.)

Theoremelrelimasn 5011 Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)

Theoremelimasn 5012 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelimasng 5013 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)

Theoremelimasni 5014 Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)

Theoremargs 5015* 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 our df-fv 4675, which was based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)

Theoremeliniseg 5016 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 5017 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoreminiseg 5018* 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 5019* 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 5020* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se

Theoremexse2 5021 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se

Theoremimass1 5022 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Theoremimass2 5023 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)

Theoremndmima 5024 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)

Theoremrelcnv 5025 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)

Theoremrelbrcnvg 5026 When is a relation, the sethood assumptions on brcnv 4838 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremeliniseg2 5027 Eliminate the class existence constraint in eliniseg 5016. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)

Theoremrelbrcnv 5028 When is a relation, the sethood assumptions on brcnv 4838 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremcotr 5029* 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 5030* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)

Theoremcnvsym 5031* 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 5032* 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 5033* Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 5171. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremasymref2 5034* 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 5035* 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 5036* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)

Theoremcodir 5037* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremqfto 5038* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)

Theoremxpidtr 5039 A square cross product is a transitive relation. (Contributed by FL, 31-Jul-2009.)

Theoremtrin2 5040 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)

Theorempoirr2 5041 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)

Theoremtrinxp 5042 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 5043 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri 5044 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremson2lpi 5045 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremsotri2 5046 A transitivity relation. (Read and implies .) (Contributed by Mario Carneiro, 10-May-2013.)

Theoremsotri3 5047 A transitivity relation. (Read and implies .) (Contributed by Mario Carneiro, 10-May-2013.)

TheoremsoirriOLD 5048 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremsotriOLD 5049 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremson2lpiOLD 5050 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorempoleloe 5051 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theorempoltletr 5052 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremsomin1 5053 Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremsomincom 5054 Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremsomin2 5055 Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremsoltmin 5056 Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremcnvopab 5057* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnv0 5058 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)

Theoremcnvi 5059 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 5060 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 5061 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)

Theoremcnvin 5062 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 5063 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Theoremrnin 5064 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremrniun 5065 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremrnuni 5066* 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 5067 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremimaundir 5068 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 5069 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 5070 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Theoremcnvxp 5071 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 5072 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 5073 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 5074 At least one member of an empty cross product is empty. (Contributed by NM, 27-Aug-2006.)

Theoremxpdisj1 5075 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpdisj2 5076 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpsndisj 5077 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)

Theoremdjudisj 5078* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremresdisj 5079 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxp 5080 The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)

Theoremdmxpss 5081 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

Theoremrnxpss 5082 The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxpid 5083 The range of a square cross product. (Contributed by FL, 17-May-2010.)

Theoremssxpb 5084 A cross-product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.)

Theoremxp11 5085 The cross product of non-empty classes is one-to-one. (Contributed by NM, 31-May-2008.)

Theoremxpcan 5086 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)

Theoremxpcan2 5087 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)

Theoremxpexr 5088 If a cross product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)

Theoremxpexr2 5089 If a nonempty cross product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)

Theoremssrnres 5090 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Theoremrninxp 5091* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdminxp 5092* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Theoremimainrect 5093 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremsossfld 5094 The base set of a strict order is contained in the field of the relation, except possibly for one element (note that ). (Contributed by Mario Carneiro, 27-Apr-2015.)

Theoremsofld 5095 The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremsoex 5096 If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)

Theoremcnvcnv3 5097* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremdfrel2 5098 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

Theoremdfrel4v 5099* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5488 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremcnvcnv 5100 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

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