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Theorem List for Metamath Proof Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrelxp 4701 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)

Theoremxpss1 4702 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpss2 4703 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpsspw 4704 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

TheoremxpsspwOLD 4705 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 4706 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)

Theoremxpexg 4707 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremxpex 4708 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremrelun 4709 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Theoremrelin1 4710 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)

Theoremrelin2 4711 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)

Theoremreldif 4712 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)

Theoremreliun 4713 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)

Theoremreliin 4714 An indexed intersection is a relation if if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremreluni 4715* 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 4716* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremrel0 4717 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)

Theoremrelopabi 4718 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)

Theoremrelopab 4719 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 4720 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 4721 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremopabid2 4722* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)

Theoreminopab 4723* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremdifopab 4724* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoreminxp 4725 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 4726 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpindir 4727 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpiindi 4728* Distributive law for cross product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremxpriindi 4729* Distributive law for cross product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremeliunxp 4730* Membership in a union of cross products. Analogue of elxp 4613 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremopeliunxp2 4731* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremraliunxp 4732* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4734, is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremrexiunxp 4733* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4735, is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremralxp 4734* 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 4735* 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 4736* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremralxpf 4737* Version of ralxp 4734 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrexxpf 4738* Version of rexxp 4735 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremiunxpf 4739* 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 4740* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2364. (Contributed by NM, 24-Feb-2014.)

Theoremrelop 4741* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.)

Theoremideqg 4742 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 4743 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)

Theoremididg 4744 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremissetid 4745 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 4746 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)

Theoremcoss2 4747 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)

Theoremcoeq1 4748 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq2 4749 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq1i 4750 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2i 4751 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq1d 4752 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2d 4753 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq12i 4754 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremcoeq12d 4755 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremnfco 4756 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)

Theorembrcog 4757* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)

Theoremopelco2g 4758* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorembrco 4759* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theoremopelco 4760* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theoremcnvss 4761 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)

Theoremcnveq 4762 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)

Theoremcnveqi 4763 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)

Theoremcnveqd 4764 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)

Theoremelcnv 4765* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)

Theoremelcnv2 4766* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)

Theoremnfcnv 4767 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremopelcnvg 4768 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theorembrcnvg 4769 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)

Theoremopelcnv 4770 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)

Theorembrcnv 4771 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)

Theoremcnvco 4772 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 4773* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)

Theoremdfdm3 4774* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Theoremdfrn2 4775* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)

Theoremdfrn3 4776* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Theoremelrn2g 4777* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremelrng 4778* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdfdm4 4779 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)

Theoremdfdmf 4780* 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 4781* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremeldm2g 4782* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremeldm 4783* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)

Theoremeldm2 4784* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)

Theoremdmss 4785 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)

Theoremdmeq 4786 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)

Theoremdmeqi 4787 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)

Theoremdmeqd 4788 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)

Theoremopeldm 4789 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)

Theorembreldm 4790 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)

Theorembreldmg 4791 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)

Theoremdmun 4792 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 4793 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremdmiun 4794 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremdmuni 4795* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)

Theoremdmopab 4796* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremdmopabss 4797* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Theoremdmopab3 4798* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Theoremdm0 4799 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 4800 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

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