Home Metamath Proof ExplorerTheorem List (p. 51 of 309) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21328) Hilbert Space Explorer (21329-22851) Users' Mathboxes (22852-30843)

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

Theoremimaundir 5001 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Theoremdminss 5002 An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.)

Theoremimainss 5003 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Theoremcnvxp 5004 The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremxp0 5005 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)

Theoremxpnz 5006 The cross product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)

Theoremxpeq0 5007 At least one member of an empty cross product is empty. (Contributed by NM, 27-Aug-2006.)

Theoremxpdisj1 5008 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpdisj2 5009 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)

Theoremxpsndisj 5010 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)

Theoremdjudisj 5011* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremresdisj 5012 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxp 5013 The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)

Theoremdmxpss 5014 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

Theoremrnxpss 5015 The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnxpid 5016 The range of a square cross product. (Contributed by FL, 17-May-2010.)

Theoremssxpb 5017 A cross-product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.)

Theoremxp11 5018 The cross product of non-empty classes is one-to-one. (Contributed by NM, 31-May-2008.)

Theoremxpcan 5019 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)

Theoremxpcan2 5020 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)

Theoremxpexr 5021 If a cross product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)

Theoremxpexr2 5022 If a nonempty cross product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)

Theoremssrnres 5023 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Theoremrninxp 5024* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdminxp 5025* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Theoremimainrect 5026 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremsossfld 5027 The base set of a strict order is contained in the field of the relation, except possibly for one element (note that ). (Contributed by Mario Carneiro, 27-Apr-2015.)

Theoremsofld 5028 The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremsoex 5029 If the relation in a strict order is a set, then the base field is also a set. (Contributed by Mario Carneiro, 27-Apr-2015.)

Theoremcnvcnv3 5030* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremdfrel2 5031 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

Theoremdfrel4v 5032* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5420 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremcnvcnv 5033 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

Theoremcnvcnv2 5034 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)

Theoremcnvcnvss 5035 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)

Theoremcnveqb 5036 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)

Theoremdfrel3 5037 Alternate definition of relation. (Contributed by NM, 14-May-2008.)

Theoremdmresv 5038 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremrnresv 5039 The range of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremdfrn4 5040 Range defined in terms of image. (Contributed by NM, 14-May-2008.)

Theoremrescnvcnv 5041 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvcnvres 5042 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)

Theoremimacnvcnv 5043 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremdmsnn0 5044 The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremrnsnn0 5045 The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.)

Theoremdmsn0 5046 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)

Theoremcnvsn0 5047 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremdmsn0el 5048 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)

Theoremrelsn2 5049 A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.)

Theoremdmsnopg 5050 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnopss 5051 The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on ). (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremdmpropg 5052 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnop 5053 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremdmprop 5054 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Theoremdmtpop 5055 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)

Theoremcnvcnvsn 5056 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5061, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnsnsn 5057 The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremrnsnopg 5058 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremrnsnop 5059 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop1sta 5060 Extract the first member of an ordered pair. (See op2nda 5063 to extract the second member, op1stb 4460 for an alternate version, and op1st 5980 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)

Theoremcnvsn 5061 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop2ndb 5062 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4460 to extract the first member, op2nda 5063 for an alternate version, and op2nd 5981 for the preferred version.) (Contributed by NM, 25-Nov-2003.)

Theoremop2nda 5063 Extract the second member of an ordered pair. (See op1sta 5060 to extract the first member, op2ndb 5062 for an alternate version, and op2nd 5981 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvsng 5064 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)

Theoremopswap 5065 Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremelxp4 5066 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5067, elxp6 6003, and elxp7 6004. (Contributed by NM, 17-Feb-2004.)

Theoremelxp5 5067 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5066 when the double intersection does not create class existence problems (caused by int0 3774). (Contributed by NM, 1-Aug-2004.)

Theoremcnvresima 5068 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Theoremresdm2 5069 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)

Theoremresdmres 5070 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremimadmres 5071 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremmptpreima 5072* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremmptiniseg 5073* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremdmmpt 5074 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremdmmptss 5075* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremdmmptg 5076* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)

Theoremrelco 5077 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Theoremdfco2 5078* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)

Theoremdfco2a 5079* Generalization of dfco2 5078, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundi 5080 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundir 5081 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcores 5082 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresco 5083 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Theoremimaco 5084 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)

Theoremrnco 5085 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)

Theoremrnco2 5086 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)

Theoremdmco 5087 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)

Theoremcoiun 5088* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)

Theoremcocnvcnv1 5089 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)

Theoremcocnvcnv2 5090 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)

Theoremcores2 5091 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)

Theoremco02 5092 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)

Theoremco01 5093 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)

Theoremcoi1 5094 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoi2 5095 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoires1 5096 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Theoremcoass 5097 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)

Theoremrelcnvtr 5098 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)

Theoremrelssdmrn 5099 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)

Theoremcnvssrndm 5100 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
 Copyright terms: Public domain < Previous  Next >