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Theorem List for Metamath Proof Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqbrrdiv 4801* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqrelrdv2 4802* A version of eqrelrdv 4799. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremssrelrel 4803* A subclass relationship determined by ordered triples. Use relrelss 5212 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremeqrelrel 4804* Extensionality principle for ordered triples (used by 2-place operations df-oprab 5878), analogous to eqrel 4793. Use relrelss 5212 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)

Theoremelrel 4805* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)

Theoremrelsn 4806 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)

Theoremrelsnop 4807 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremxpss12 4808 Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremxpss 4809 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrelxp 4810 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)

Theoremxpss1 4811 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpss2 4812 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpsspw 4813 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

TheoremxpsspwOLD 4814 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremunixpss 4815 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)

Theoremxpexg 4816 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. See also xpexgALT 6086. (Contributed by NM, 14-Aug-1994.)

Theoremxpex 4817 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremrelun 4818 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Theoremrelin1 4819 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)

Theoremrelin2 4820 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)

Theoremreldif 4821 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)

Theoremreliun 4822 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)

Theoremreliin 4823 An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremreluni 4824* The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)

Theoremrelint 4825* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremrel0 4826 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)

Theoremrelopabi 4827 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)

Theoremrelopab 4828 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 4829 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 4830 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremopabid2 4831* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)

Theoreminopab 4832* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremdifopab 4833* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoreminxp 4834 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 4835 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpindir 4836 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpiindi 4837* Distributive law for cross product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremxpriindi 4838* Distributive law for cross product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremeliunxp 4839* Membership in a union of cross products. Analogue of elxp 4722 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremopeliunxp2 4840* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremraliunxp 4841* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4843, is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremrexiunxp 4842* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4844, is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremralxp 4843* 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 4844* 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 4845* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremralxpf 4846* Version of ralxp 4843 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrexxpf 4847* Version of rexxp 4844 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremiunxpf 4848* 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 4849* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2411. (Contributed by NM, 24-Feb-2014.)

Theoremrelop 4850* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.)

Theoremideqg 4851 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 4852 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)

Theoremididg 4853 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremissetid 4854 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 4855 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)

Theoremcoss2 4856 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)

Theoremcoeq1 4857 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq2 4858 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq1i 4859 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2i 4860 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq1d 4861 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2d 4862 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq12i 4863 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremcoeq12d 4864 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremnfco 4865 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)

Theorembrcog 4866* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)

Theoremopelco2g 4867* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorembrco 4868* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theoremopelco 4869* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theoremcnvss 4870 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)

Theoremcnveq 4871 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)

Theoremcnveqi 4872 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)

Theoremcnveqd 4873 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)

Theoremelcnv 4874* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)

Theoremelcnv2 4875* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)

Theoremnfcnv 4876 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremopelcnvg 4877 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theorembrcnvg 4878 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)

Theoremopelcnv 4879 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)

Theorembrcnv 4880 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)

Theoremcnvco 4881 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 4882* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)

Theoremdfdm3 4883* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Theoremdfrn2 4884* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)

Theoremdfrn3 4885* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Theoremelrn2g 4886* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremelrng 4887* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdfdm4 4888 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)

Theoremdfdmf 4889* 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 4890* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremeldm2g 4891* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremeldm 4892* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)

Theoremeldm2 4893* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)

Theoremdmss 4894 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)

Theoremdmeq 4895 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)

Theoremdmeqi 4896 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)

Theoremdmeqd 4897 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)

Theoremopeldm 4898 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)

Theorembreldm 4899 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)

Theorembreldmg 4900 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)

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