Home Metamath Proof ExplorerTheorem List (p. 253 of 328) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21514) Hilbert Space Explorer (21515-23037) Users' Mathboxes (23038-32776)

Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremimrestr 25201 Image of an element of transitive class under a class restricted by . (Contributed by FL, 20-Mar-2011.)

Theoremimresord 25202 Image of an element of an ordinal under a class restricted by . (Contributed by FL, 20-Mar-2011.)

Theoremsndw 25203 If is a part of and a part of and is equipotent to then is equipotent to . The art of sandwich applied to set theory. (Contributed by FL, 16-Apr-2011.) (Revised by Mario Carneiro, 3-May-2015.)

Theoremsndw2 25204 If is a part of and a part of and is equipotent to then is equipotent to . The art of sandwich applied to set theory. (Contributed by FL, 16-Apr-2011.)

Theoremordsuccl 25205 If a successor of belongs to an ordinal, so does . (Contributed by FL, 20-Mar-2011.)

Theoremordsuccl2 25206 If a successor of belongs to an ordinal, so does . (Contributed by FL, 20-Mar-2011.)

Theoremordsuccl3 25207 If a successor of belongs to an ordinal, is a part of the ordinal. (Contributed by FL, 20-Mar-2011.)

Theoremdomtri3 25208 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by FL, 16-Apr-2011.)

Theoremisfinite1b 25209 Omega strictly dominates a finite set. (Contributed by FL, 16-Apr-2011.)

Theoremcptwff 25210 The cross product of two finite sets is finite. (Contributed by FL, 16-Apr-2011.)

Theoreminttrp 25211 The intersection of a non-empty element of a transitive class is a part of the class. (Contributed by FL, 15-Apr-2011.)

Theoremtrunitr 25212 The union of two transitive classes is transitive. JFM CLASSES1. th. 55 (Contributed by FL, 16-Apr-2011.)

Theoremuncum2 25213* Union of a cumulative hierarchy of sets. (Contributed by FL, 23-Apr-2011.)

Theoremcelsor 25214* If all the elements of a set are ordinal numbers and are parts of the set then is an ordinal number. (Contributed by FL, 20-Apr-2011.)

Theoremreflincror 25215 If a relation is reflexive, it is included in . (Contributed by FL, 8-May-2011.)

Theoremfldrels 25216 The field of a relation is a set. (Contributed by FL, 23-May-2011.)

Theoremfvsnn 25217 Value when doesn't belong to the domain. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 3-May-2015.)

Theoremfvsn2a 25218 Value of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.)

Theoremfvsn2b 25219 Value of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.)

Theoremrelrefcnv 25220 A relation is reflexive iff its converse is reflexive. (Contributed by FL, 19-Sep-2011.)

Theoremeqfnung2 25221* If a family of sets indexed by covers the common domain of two functions and , the restrictions of and to are equal iff . Compare eqfnun 26490. (Contributed by FL, 5-Nov-2011.)

Theoreminjrec2 25222* A function is an injection iff a retraction exists. Bourbaki E.II.18 prop. 8. (Contributed by FL, 11-Nov-2011.)

Theoremsurjsec2 25223* A function is a surjection iff a section exists. Bourbaki E.II.18 prop. 8. (Contributed by FL, 18-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremab2rexexg2 25224* Existence of a class abstraction of existentially restricted sets. (Contributed by FL, 15-Oct-2012.)

Theoremab2rexexg 25225* Existence of a class abstraction of existentially restricted sets. (Contributed by FL, 19-Apr-2012.)

Theoremov2gc 25226* Value of a composition. ovmpt2g 5998 adapted to this special case of a composite. (Contributed by FL, 14-Jul-2012.)

Theoremov4gc 25227* Value of a composition. ovmpt4g 5986 adapted to the special case of a composite. (Contributed by FL, 14-Jul-2012.)

Theoremdomintrefc 25228* The domain of the intersection of a family of reflexive classes is the intersection of the domains. (Contributed by FL, 15-Oct-2012.)

Theoremrnintintrn 25229* The range of an intersection is a part of the intersection of the ranges. (The case works as well, the intersection gives ). (Contributed by FL, 15-Oct-2012.)

Theoremprjpacp1 25230 Projection of a part of a cross product. (Contributed by FL, 15-Oct-2012.)

Theoremprjpacp2 25231 Projection of a part of a cross product. (Contributed by FL, 15-Oct-2012.)

Theoremrelinccppr 25232 A relation is included in the cross product of its projections. (Contributed by FL, 15-Oct-2012.)

Theoremdffn5a 25233* Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theoremffvelrnb 25234 A function's value belongs to its codomain. (Contributed by FL, 14-Sep-2013.)

Theoremab2rexex2g 25235* Existence of an existentially restricted class abstraction. is normally has free-variable parameters , , and . Compare abrexex2g 5784. (Contributed by FL, 6-Nov-2013.)

Theoremfprg 25236 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)

Theoremiccss3 25237 Condition for a closed interval to be a subset of another closed interval. See iccss (Contributed by FL, 29-May-2014.)

Theoremiccleub2 25238 An element of a closed interval is more than or equal to its lower bound. (Contributed by FL, 29-May-2014.)

Theoremiccleub3 25239 An element of a closed interval is less than or equal to its upper bound. (Contributed by FL, 29-May-2014.)

Theoremxrre3OLD 25240 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.) (Moved into main set.mm as xrre3 10516 and may be deleted by mathbox owner, FL. --NM 26-Aug-2017.)

Theoreminabs2 25241 Absorption law for intersection. (Contributed by FL, 30-May-2014.)

Theoreminttpemp 25242 Two ways of saying that two triples have no common element. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Theoremmapex2 25243* Two ways to express a subset of mappings. (Contributed by FL, 17-Nov-2014.)

Theoremsssu 25244 Equality of a class difference and it subtrahend. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)

18.13.6  The "maps to" notation

TheoremcmpfunOLD 25245 Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) (Moved into main set.mm as funmpt2 5307 and may be deleted by mathbox owner, FL. --NM 26-Aug-2017.)

Theoremcmpdom 25246* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-Feb-2008.)

Theoremcmpdom2 25247* Domain of a class given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)

Theoremfopab2g 25248* Functionality of an ordered-pair class abstraction given by the "maps to" notation. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 31-May-2014.)

Theoremcrimmt1 25249* Composition of a restricted identity and a mapping (using the maps to notation). See fcoi1 5431. (Contributed by FL, 25-Apr-2012.)

Theoremcrimmt2 25250* Composition of a restricted identity and a mapping (using the maps to notation). See fcoi2 5432. (Contributed by FL, 25-Apr-2012.)

Theoremmapmapmap 25251* Function returning a composite. (Contributed by FL, 19-Nov-2011.)

Theoreminjsurinj 25252* If is an injection and a surjection is an injection. Bourbaki E.II.31 prop. 2. (Contributed by FL, 20-Nov-2011.)

18.13.7  Cartesian Products

In what follows I will call nuple an element of a cartesian product.

If is a cartesian product, a nuple of , an indice, is the th coordinate of the nuple .

Suppose the set of indices is and is the cartesian product then and .

Syntaxcpro 25253 Extend class notation to include the projection mapping.

Syntaxcproj 25254 Extend class notation to include the projection mapping.

Definitiondf-pro 25255* Definition of a projection (also called a coordinate mapping). Meaninful if is a cartesian product and is an index. (Contributed by FL, 19-Jun-2011.)

Definitiondf-prj 25256* Definition of a projection. Meaninful if is a cartesian product and is a set of indices. Suppose then and . (Contributed by FL, 19-Jun-2011.)

Theoremelixp2b 25257* The base class of the elements of a nuple. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2016.)

Theorembclelnu 25258* The base class of an element of a nuple. (Contributed by FL, 19-Jun-2011.)

Theoremispr1 25259* Definition of the coordinate mapping (or projection ) of index . is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremprmapcp2 25260* A projection is a mapping from a cartesian product to an element of the family implied in the product. Bourbaki E.II.34 cor. 1. (Contributed by FL, 19-Jun-2011.)

Theoremvalpr 25261 The th coordinate of the nuple . (Contributed by FL, 19-Jun-2011.)

Theoremnpincppr 25262* A set of nuples is included in the cartesian product of the projections of the nuples. Bourbaki E.II.32. (Contributed by FL, 20-Jun-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)

Theoremrepfuntw 25263 Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.)

Theoremrepcpwti 25264* A representation of a cartesian product with two indices. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Theoremcbcpcp 25265* The canonical bijection between a cross product and a cartesian product (whose set of indices is composed of two different elements). Bourbaki E.II.33 . (Contributed by FL, 26-Jun-2011.)

Theoremisprj1 25266* Definition of a projection. is a set of indices. is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremisprj2 25267* Definition of a projection. is a set of indices. is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)

Theoremprjmapcp 25268* A projection is a mapping from a cartesian product to one of its restriction. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)

Theoremcbicp 25269* Canonical bijection between a cartesian product indexed by a singleton and the base class of the elements of the 1-uple. Bourbaki E II.32 (Contributed by FL, 6-Jun-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)

Theoremprl 25270* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 7-Nov-2011.)

Theoremprl1 25271* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 20-Nov-2011.)

Theoremprl2 25272* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 20-Nov-2011.)

Theoremprjmapcp2 25273* A projection is a mapping from a cartesian product onto one of its restriction. Bourbaki E.II.33 prop. 5. (Contributed by FL, 20-Nov-2011.) (Revised by Mario Carneiro, 31-May-2014.)

Theoremdstr 25274* Distribution of union over intersection. Bourbaki E.II.35 prop. 8. (Contributed by FL, 18-Jun-2011.)

18.13.8  Operations on subsets and functions

Syntaxccst 25275 Extend class notation with an operator that derives an operation on subsets of a set from an operation on the elements of this set.

Definitiondf-cst 25276* Define an operation on the subsets derived from an operation on the elements. Meaningful if is a binary internal operation. (Contributed by FL, 18-Apr-2010.)

Theoremiscst1 25277* An operation on the subsets derived from an operation on the elements. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremiscst2 25278* The value of the couple through the derived operation. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremiscst3 25279* Property equivalent to the fact of belonging to the value of a pair through the derived operation. (Contributed by FL, 18-Apr-2010.)

Theoremiscst4 25280* The value of the couple through the derived operation (expressed with a union). (Contributed by FL, 31-Dec-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)

18.13.9  Arithmetic

Theorem3timesi 25281 Three times a number. (Contributed by FL, 17-Oct-2010.)

Theorem2eq3m1 25282 equals minus . (Contributed by FL, 17-Oct-2010.)

TheoremnZdef 25283* Two ways to define . In the first way I multiply the set by the set ( I think this is this sort of multiplication that is at the origin of the denotation ). In the second way I multiply the integer by an element of . (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)

18.13.10  Lattice (algebraic definition)

Syntaxclatalg 25284 Extend class notation to include the class of lattices.

Definitiondf-latalg 25285* Algebraic definition of a lattice. is called the join of the lattice (and is denoted by in textbooks) , is called the meet (and is denoted by in textbooks). (Contributed by FL, 11-Dec-2009.)

Theoremislatalg 25286* The predicate "is a lattice". (Contributed by FL, 11-Dec-2009.)

Theoremjop 25287 Join is a binary internal operation. (Contributed by FL, 12-Dec-2009.)

Theoremmop 25288 Meet is a binary internal operation. (Contributed by FL, 12-Dec-2009.)

Theoremcljo 25289 Closure of join. (Contributed by FL, 12-Dec-2009.)

Theoremclme 25290 Closure of meet. (Contributed by FL, 12-Dec-2009.)

Theoremlabs1 25291* Absorption law. . (Contributed by FL, 12-Dec-2009.)

Theoremlabss1 25292 Absorption law. . (Contributed by FL, 12-Dec-2009.)

Theoremlabs2 25293* Absorption law. . (Contributed by FL, 12-Dec-2009.)

Theoremlabss2 25294 Absorption law. . (Contributed by FL, 12-Dec-2009.)

Theoremjidd 25295 Join is idempotent. . (Contributed by FL, 12-Dec-2009.)

Theoremmidd 25296 Meet is idempotent. . (Contributed by FL, 12-Dec-2009.)

18.13.11  Currying and Partial Mappings

Syntaxccur1 25297 Extend class notation with the definition of currying.

Syntaxccur2 25298 Extend class notation with the definition of currying.

Definitiondf-cur1 25299* Definition of currying (1st sort). Currying the operation consists in creating a mapping that returns for every value of the partial application of to . (Contributed by FL, 24-Jan-2010.)

Definitiondf-cur2 25300* Definition of currying (2nd sort). Currying the operation consists in creating a mapping that returns for every value of the partial application of to . (Contributed by FL, 24-Jan-2010.)

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 32701-32776
 Copyright terms: Public domain < Previous  Next >