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

Theoremdmv 4801 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)

Theoremdm0rn0 4802 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)

Theoremreldm0 4803 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)

Theoremdmxp 4804 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 4805 The domain of a square cross product. (Contributed by NM, 28-Jul-1995.)

Theoremdmxpin 4806 The domain of the intersection of two square cross products. Unlike dmin 4793, equality holds. (Contributed by NM, 29-Jan-2008.)

Theoremxpid11 4807 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 4808 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5031). (Contributed by NM, 8-Apr-2007.)

Theoremrncnvcnv 4809 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremelreldm 4810 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)

Theoremrneq 4811 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)

Theoremrneqi 4812 Equality inference for range. (Contributed by NM, 4-Mar-2004.)

Theoremrneqd 4813 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)

Theoremrnss 4814 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)

Theorembrelrng 4815 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)

Theorembrelrn 4816 The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)

Theoremopelrn 4817 Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)

Theoremreleldm 4818 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)

Theoremrelelrn 4819 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)

Theoremreleldmb 4820* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)

Theoremrelelrnb 4821* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)

Theoremreleldmi 4822 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)

Theoremrelelrni 4823 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)

Theoremdfrnf 4824* Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremelrn2 4825* Membership in a range. (Contributed by NM, 10-Jul-1994.)

Theoremelrn 4826* Membership in a range. (Contributed by NM, 2-Apr-2004.)

Theoremnfdm 4827 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfrn 4828 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremdmiin 4829 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)

Theoremcsbrng 4830 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremrnopab 4831* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremrnmpt 4832* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpt 4833* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremelrnmpt1s 4834* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremelrnmpt1 4835 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmptg 4836* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpti 4837* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfiun3g 4838 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3g 4839 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiun3 4840 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3 4841 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremriinint 4842* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremrn0 4843 The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.)

Theoremrelrn0 4844 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)

Theoremdmrnssfld 4845 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)

Theoremdmexg 4846 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)

Theoremrnexg 4847 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)

Theoremdmex 4848 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)

Theoremrnex 4849 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)

Theoremiprc 4850 The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn 16819. (Contributed by NM, 1-Jan-2007.)

Theoremdmcoss 4851 Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrncoss 4852 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremdmcosseq 4853 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmcoeq 4854 Domain of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremrncoeq 4855 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremreseq1 4856 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)

Theoremreseq2 4857 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)

Theoremreseq1i 4858 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq2i 4859 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12i 4860 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq1d 4861 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq2d 4862 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12d 4863 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremnfres 4864 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremcsbresg 4865 Distribute proper substitution through the restriction of a class. csbresg 4865 is derived from the virtual deduction proof csbresgVD 27361. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremres0 4866 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)

Theoremopelres 4867 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)

Theorembrres 4868 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)

Theoremopelresg 4869 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)

Theorembrresg 4870 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)

Theoremopres 4871 Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresieq 4872 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)

TheoremopelresiOLD 4873 belongs to a restriction of the identity class iff belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremopelresi 4874 belongs to a restriction of the identity class iff belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)

Theoremresres 4875 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)

Theoremresundi 4876 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremresundir 4877 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)

Theoremresindi 4878 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)

Theoremresindir 4879 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)

Theoreminres 4880 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)

Theoremresiun1 4881* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremresiun2 4882* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremdmres 4883 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)

Theoremssdmres 4884 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)

Theoremdmresexg 4885 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)

Theoremresss 4886 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrescom 4887 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremssres 4888 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremssres2 4889 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrelres 4890 A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresabs1 4891 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)

Theoremresabs2 4892 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresidm 4893 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresima 4894 A restriction to an image. (Contributed by NM, 29-Sep-2004.)

Theoremresima2 4895 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)

Theoremxpssres 4896 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremelres 4897* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremelsnres 4898* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremrelssres 4899 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremresdm 4900 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)

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