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

Theoremrel0 5001 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)

Theoremrelopabi 5002 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)

Theoremrelopab 5003 A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)

Theoremreli 5004 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremrele 5005 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremopabid2 5006* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)

Theoreminopab 5007* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremdifopab 5008* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoreminxp 5009 The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremxpindi 5010 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpindir 5011 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpiindi 5012* Distributive law for cross product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremxpriindi 5013* Distributive law for cross product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremeliunxp 5014* Membership in a union of cross products. Analogue of elxp 4897 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremopeliunxp2 5015* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremraliunxp 5016* Write a double restricted quantification as one universal quantifier. In this version of ralxp 5018, is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremrexiunxp 5017* Write a double restricted quantification as one universal quantifier. In this version of rexxp 5019, is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremralxp 5018* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremrexxp 5019* Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)

Theoremdjussxp 5020* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremralxpf 5021* Version of ralxp 5018 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrexxpf 5022* Version of rexxp 5019 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremiunxpf 5023* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)

Theoremopabbi2dv 5024* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2553. (Contributed by NM, 24-Feb-2014.)

Theoremrelop 5025* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.)

Theoremideqg 5026 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremideq 5027 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)

Theoremididg 5028 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremissetid 5029 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremcoss1 5030 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)

Theoremcoss2 5031 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)

Theoremcoeq1 5032 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq2 5033 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq1i 5034 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2i 5035 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq1d 5036 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2d 5037 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq12i 5038 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremcoeq12d 5039 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremnfco 5040 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)

Theorembrcog 5041* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)

Theoremopelco2g 5042* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorembrcogw 5043 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Theoremeqbrrdva 5044* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theorembrco 5045* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theoremopelco 5046* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theoremcnvss 5047 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)

Theoremcnveq 5048 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)

Theoremcnveqi 5049 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)

Theoremcnveqd 5050 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)

Theoremelcnv 5051* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)

Theoremelcnv2 5052* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)

Theoremnfcnv 5053 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremopelcnvg 5054 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theorembrcnvg 5055 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)

Theoremopelcnv 5056 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)

Theorembrcnv 5057 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)

Theoremcnvco 5058 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvuni 5059* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)

Theoremdfdm3 5060* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Theoremdfrn2 5061* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)

Theoremdfrn3 5062* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Theoremelrn2g 5063* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremelrng 5064* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdfdm4 5065 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)

Theoremdfdmf 5066* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremeldmg 5067* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremeldm2g 5068* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremeldm 5069* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)

Theoremeldm2 5070* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)

Theoremdmss 5071 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)

Theoremdmeq 5072 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)

Theoremdmeqi 5073 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)

Theoremdmeqd 5074 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)

Theoremopeldm 5075 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)

Theorembreldm 5076 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)

Theorembreldmg 5077 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)

Theoremdmun 5078 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmin 5079 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremdmiun 5080 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremdmuni 5081* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)

Theoremdmopab 5082* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremdmopabss 5083* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Theoremdmopab3 5084* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Theoremdm0 5085 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmi 5086 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmv 5087 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)

Theoremdm0rn0 5088 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)

Theoremreldm0 5089 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)

Theoremdmxp 5090 The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmxpid 5091 The domain of a square cross product. (Contributed by NM, 28-Jul-1995.)

Theoremdmxpin 5092 The domain of the intersection of two square cross products. Unlike dmin 5079, equality holds. (Contributed by NM, 29-Jan-2008.)

Theoremxpid11 5093 The cross product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmcnvcnv 5094 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5323). (Contributed by NM, 8-Apr-2007.)

Theoremrncnvcnv 5095 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremelreldm 5096 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)

Theoremrneq 5097 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)

Theoremrneqi 5098 Equality inference for range. (Contributed by NM, 4-Mar-2004.)

Theoremrneqd 5099 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)

Theoremrnss 5100 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)

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