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Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelrn 5101* Membership in a range. (Contributed by NM, 2-Apr-2004.)

Theoremnfdm 5102 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfrn 5103 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremdmiin 5104 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)

Theoremcsbrng 5105 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremrnopab 5106* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremrnmpt 5107* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpt 5108* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremelrnmpt1s 5109* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremelrnmpt1 5110 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmptg 5111* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpti 5112* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfiun3g 5113 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3g 5114 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiun3 5115 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3 5116 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremriinint 5117* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremrn0 5118 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 5119 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)

Theoremdmrnssfld 5120 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)

Theoremdmexg 5121 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)

Theoremrnexg 5122 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 5123 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)

Theoremrnex 5124 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 5125 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 17309. (Contributed by NM, 1-Jan-2007.)

Theoremdmcoss 5126 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 5127 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremdmcosseq 5128 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmcoeq 5129 Domain of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremrncoeq 5130 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremreseq1 5131 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)

Theoremreseq2 5132 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)

Theoremreseq1i 5133 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq2i 5134 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12i 5135 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq1d 5136 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq2d 5137 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12d 5138 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremnfres 5139 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremcsbresg 5140 Distribute proper substitution through the restriction of a class. csbresg 5140 is derived from the virtual deduction proof csbresgVD 28861. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremres0 5141 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)

Theoremopelres 5142 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)

Theorembrres 5143 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)

Theoremopelresg 5144 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)

Theorembrresg 5145 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)

Theoremopres 5146 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 5147 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)

TheoremopelresiOLD 5148 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 5149 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 5150 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)

Theoremresundi 5151 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremresundir 5152 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)

Theoremresindi 5153 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)

Theoremresindir 5154 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)

Theoreminres 5155 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)

Theoremresiun1 5156* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremresiun2 5157* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremdmres 5158 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)

Theoremssdmres 5159 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)

Theoremdmresexg 5160 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)

Theoremresss 5161 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrescom 5162 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremssres 5163 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremssres2 5164 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrelres 5165 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 5166 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)

Theoremresabs2 5167 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresidm 5168 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresima 5169 A restriction to an image. (Contributed by NM, 29-Sep-2004.)

Theoremresima2 5170 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)

Theoremxpssres 5171 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremelres 5172* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremelsnres 5173* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremrelssres 5174 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremresdm 5175 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)

Theoremresexg 5176 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresex 5177 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremresopab 5178* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)

Theoremresiexg 5179 The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5948). (Contributed by NM, 13-Jan-2007.)

Theoremiss 5180 A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresopab2 5181* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)

Theoremresmpt 5182* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)

Theoremresmpt3 5183* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)

Theoremdfres2 5184* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremopabresid 5185* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

Theoremmptresid 5186* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)

Theoremdmresi 5187 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresid 5188 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)

Theoremimaeq1 5189 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq2 5190 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq1i 5191 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq2i 5192 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq1d 5193 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq2d 5194 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq12d 5195 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)

Theoremdfima2 5196* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdfima3 5197* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremelimag 5198* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)

Theoremelima 5199* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)

Theoremelima2 5200* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)

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