Home Metamath Proof ExplorerTheorem List (p. 64 of 329) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22423) Hilbert Space Explorer (22424-23946) Users' Mathboxes (23947-32824)

Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremf1opw 6301* A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremsuppss2 6302* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.)

Theoremsuppssfv 6303* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremsuppssov1 6304* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)

2.4.12  Function operation

Syntaxcof 6305 Extend class notation to include mapping of an operation to a function operation.

Syntaxcofr 6306 Extend class notation to include mapping of a binary relation to a function relation.

Definitiondf-of 6307* Define the function operation map. The definition is designed so that if is a binary operation, then is the analogous operation on functions which corresponds to applying pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)

Definitiondf-ofr 6308* Define the function relation map. The definition is designed so that if is a binary relation, then is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofeq 6309 Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofreq 6310 Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofexg 6311 A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)

Theoremnfof 6312* Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremnfofr 6313* Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremoffval 6314* Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrfval 6315* Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremofval 6316 Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrval 6317 Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremoffn 6318 The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremfnfvof 6319 Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)

Theoremoffval3 6320* General value of with no assumptions on functionality of and . (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremoffres 6321 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremoff 6322* The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofres 6323 Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremoffval2 6324* The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofrfval2 6325* The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)

Theoremofco 6326 The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)

Theoremoffveq 6327* Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremoffveqb 6328* Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremofc1 6329 Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofc2 6330 Right operation by a constant. (Contributed by NM, 7-Oct-2014.)

Theoremofc12 6331 Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofref 6332* Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofinvl 6333* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)

Theoremcaofid0l 6334* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Theoremcaofid0r 6335* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Theoremcaofid1 6336* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofid2 6337* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofcom 6338* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaofrss 6339* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofass 6340* Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaoftrn 6341* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Theoremcaofdi 6342* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremcaofdir 6343* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)

Theoremcaonncan 6344* Transfer nncan 9332-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Theoremofmres 6345* Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, can be a set by ofmresex 6347, allowing it to be used as a function or structure argument. By ofmresval 6346, the restricted operation map values are the same as the original values, allowing theorems for to be reused. (Contributed by NM, 20-Oct-2014.)

Theoremofmresval 6346 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Theoremofmresex 6347 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Theoremsuppssof1 6348* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)

2.4.13  First and second members of an ordered pair

Syntaxc1st 6349 Extend the definition of a class to include the first member an ordered pair function.

Syntaxc2nd 6350 Extend the definition of a class to include the second member an ordered pair function.

Definitiondf-1st 6351 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6357 proves that it does this. For example, . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5353 and op1stb 4760). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Definitiondf-2nd 6352 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6358 proves that it does this. For example, . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5356 and op2ndb 5355). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Theorem1stval 6353 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theorem2ndval 6354 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theorem1st0 6355 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Theorem2nd0 6356 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Theoremop1st 6357 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop2nd 6358 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop1std 6359 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop2ndd 6360 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop1stg 6361 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremop2ndg 6362 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremot1stg 6363 Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6363, ot2ndg 6364, ot3rdg 6365.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot2ndg 6364 Extract the second member of an ordered triple. (See ot1stg 6363 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot3rdg 6365 Extract the third member of an ordered triple. (See ot1stg 6363 comment.) (Contributed by NM, 3-Apr-2015.)

Theorem1stval2 6366 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Theorem2ndval2 6367 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Theoremfo1st 6368 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo2nd 6369 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf1stres 6370 Mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf2ndres 6371 Mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo1stres 6372 Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Theoremfo2ndres 6373 Onto mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Theorem1st2val 6374* Value of an alternate definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theorem2nd2val 6375* Value of an alternate definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theorem1stcof 6376 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)

Theorem2ndcof 6377 Composition of the first member function with another function. (Contributed by FL, 15-Oct-2012.)

Theoremxp1st 6378 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremxp2nd 6379 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremelxp6 6380 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5359. (Contributed by NM, 9-Oct-2004.)

Theoremelxp7 6381 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5359. (Contributed by NM, 19-Aug-2006.)

Theoremdifxp 6382 Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremdifxp1 6383 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremdifxp2 6384 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremeqopi 6385 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)

Theoremxp2 6386* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)

Theoremunielxp 6387 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)

Theorem1st2nd2 6388 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)

Theorem1st2ndb 6389 Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)

Theoremxpopth 6390 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)

Theoremeqop 6391 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Theoremeqop2 6392 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)

Theoremop1steq 6393* Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)

Theorem2nd1st 6394 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)

Theorem1st2nd 6395 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)

Theorem1stdm 6396 The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)

Theorem2ndrn 6397 The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)

Theorem1st2ndbr 6398 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremreleldm2 6399* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremreldm 6400* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)

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 32701-32800 329 32801-32824
 Copyright terms: Public domain < Previous  Next >