Home Metamath Proof ExplorerTheorem List (p. 54 of 328) < Previous  Next > Browser slow? Try the Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-22421) Hilbert Space Explorer (22422-23944) Users' Mathboxes (23945-32762)

Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrnxpss 5301 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 5302 The range of a square cross product. (Contributed by FL, 17-May-2010.)

Theoremssxpb 5303 A cross-product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.)

Theoremxp11 5304 The cross product of non-empty classes is one-to-one. (Contributed by NM, 31-May-2008.)

Theoremxpcan 5305 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)

Theoremxpcan2 5306 Cancellation law for cross-product. (Contributed by NM, 30-Aug-2011.)

Theoremxpexr 5307 If a cross product is a set, one of its components must be a set. (Contributed by NM, 27-Aug-2006.)

Theoremxpexr2 5308 If a nonempty cross product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.)

Theoremssrnres 5309 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)

Theoremrninxp 5310* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdminxp 5311* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)

Theoremimainrect 5312 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremxpima 5313 The image by a constant function (or other cross product). (Contributed by Thierry Arnoux, 4-Feb-2017.)

Theoremxpima1 5314 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremxpima2 5315 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Theoremxpimasn 5316 The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Theoremsossfld 5317 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 5318 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 5319 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 5320* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremdfrel2 5321 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

Theoremdfrel4v 5322* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5772 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)

Theoremcnvcnv 5323 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)

Theoremcnvcnv2 5324 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)

Theoremcnvcnvss 5325 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)

Theoremcnveqb 5326 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)

Theoremcnveq0 5327 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)

Theoremdfrel3 5328 Alternate definition of relation. (Contributed by NM, 14-May-2008.)

Theoremdmresv 5329 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremrnresv 5330 The range of a universal restriction. (Contributed by NM, 14-May-2008.)

Theoremdfrn4 5331 Range defined in terms of image. (Contributed by NM, 14-May-2008.)

Theoremrescnvcnv 5332 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvcnvres 5333 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)

Theoremimacnvcnv 5334 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremdmsnn0 5335 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 5336 The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.)

Theoremdmsn0 5337 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)

Theoremcnvsn0 5338 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremdmsn0el 5339 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)

Theoremrelsn2 5340 A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.)

Theoremdmsnopg 5341 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnopss 5342 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 5343 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnop 5344 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 5345 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Theoremdmtpop 5346 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)

Theoremcnvcnvsn 5347 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5352, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdmsnsnsn 5348 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 5349 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 5350 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 5351 Extract the first member of an ordered pair. (See op2nda 5354 to extract the second member, op1stb 4758 for an alternate version, and op1st 6355 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)

Theoremcnvsn 5352 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremop2ndb 5353 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4758 to extract the first member, op2nda 5354 for an alternate version, and op2nd 6356 for the preferred version.) (Contributed by NM, 25-Nov-2003.)

Theoremop2nda 5354 Extract the second member of an ordered pair. (See op1sta 5351 to extract the first member, op2ndb 5353 for an alternate version, and op2nd 6356 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvsng 5355 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)

Theoremopswap 5356 Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)

Theoremelxp4 5357 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5358, elxp6 6378, and elxp7 6379. (Contributed by NM, 17-Feb-2004.)

Theoremelxp5 5358 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5357 when the double intersection does not create class existence problems (caused by int0 4064). (Contributed by NM, 1-Aug-2004.)

Theoremcnvresima 5359 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Theoremresdm2 5360 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)

Theoremresdmres 5361 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremimadmres 5362 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Theoremmptpreima 5363* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremmptiniseg 5364* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Theoremdmmpt 5365 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremdmmptss 5366* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)

Theoremdmmptg 5367* 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 5368 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)

Theoremdfco2 5369* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)

Theoremdfco2a 5370* Generalization of dfco2 5369, where can have any value between and . (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundi 5371 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcoundir 5372 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcores 5373 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresco 5374 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Theoremimaco 5375 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)

Theoremrnco 5376 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)

Theoremrnco2 5377 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)

Theoremdmco 5378 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)

Theoremcoiun 5379* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)

Theoremcocnvcnv1 5380 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)

Theoremcocnvcnv2 5381 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)

Theoremcores2 5382 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)

Theoremco02 5383 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)

Theoremco01 5384 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)

Theoremcoi1 5385 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoi2 5386 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Theoremcoires1 5387 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Theoremcoass 5388 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 5389 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)

Theoremrelssdmrn 5390 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 5391 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcossxp 5392 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)

Theoremrelrelss 5393 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)

Theoremunielrel 5394 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)

Theoremrelfld 5395 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)

Theoremrelresfld 5396 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)

Theoremrelcoi2 5397 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)

Theoremrelcoi1 5398 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)

Theoremunidmrn 5399 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)

Theoremrelcnvfld 5400 if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

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 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32762
 Copyright terms: Public domain < Previous  Next >