Home Metamath Proof ExplorerTheorem List (p. 62 of 330) < 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 (1-22459) Hilbert Space Explorer (22460-23982) Users' Mathboxes (23983-32936)

Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisopolem 6101 Lemma for isopo 6102. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisopo 6102 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisosolem 6103 Lemma for isoso 6104. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisoso 6104 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisowe 6105 An isomorphism preserves well-ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremisowe2 6106* A weak form of isowe 6105 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremf1oiso 6107* Any one-to-one onto function determines an isomorphism with an induced relation . Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)

Theoremf1oiso2 6108* Any one-to-one onto function determines an isomorphism with an induced relation . (Contributed by Mario Carneiro, 9-Mar-2013.)

Theoremf1owe 6109* Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)

Theoremf1oweALT 6110* Well-ordering of isomorphic relations. (This version is proved directly instead of with the isomorphism predicate.) (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremweniso 6111 A set-like well-ordering has no nontrivial automorphisms. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
Se

Theoremweisoeq 6112 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 6114. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se

Theoremweisoeq2 6113 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 6115. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se

Theoremwemoiso 6114* Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 8032. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremwemoiso2 6115* Thus, there is at most one isomorphism between any two well-ordered sets. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremknatar 6116* The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice . (Contributed by Mario Carneiro, 11-Jun-2015.)

2.4.10  Operations

Syntaxco 6117 Extend class notation to include the value of an operation (such as ) for two arguments and . Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 9330.)

Syntaxcoprab 6118 Extend class notation to include class abstraction (class builder) of nested ordered pairs.

Syntaxcmpt2 6119 Extend the definition of a class to include maps-to notation for defining an operation via a rule.

Definitiondf-ov 6120 Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation and its arguments and - will be useful for proving meaningful theorems. For example, if class is the operation and arguments and are and , the expression can be proved to equal (see 3p2e5 10149). This definition is well-defined, although not very meaningful, when classes and/or are proper classes (i.e. are not sets); see ovprc1 6145 and ovprc2 6146. On the other hand, we often find uses for this definition when is a proper class, such as in oav 6791. is normally equal to a class of nested ordered pairs of the form defined by df-oprab 6121. (Contributed by NM, 28-Feb-1995.)

Definitiondf-oprab 6121* Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally , , and are distinct, although the definition doesn't strictly require it. See df-ov 6120 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ovmpt2 6245. (Contributed by NM, 12-Mar-1995.)

Definitiondf-mpt2 6122* Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from (in ) to ." An extension of df-mpt 4299 for two arguments. (Contributed by NM, 17-Feb-2008.)

Theoremoveq 6123 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq1 6124 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq2 6125 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq12 6126 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)

Theoremoveq1i 6127 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq2i 6128 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)

Theoremoveq12i 6129 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoveqi 6130 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)

Theoremoveq123i 6131 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)

Theoremoveq1d 6132 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)

Theoremoveq2d 6133 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)

Theoremoveqd 6134 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)

Theoremoveq12d 6135 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremoveqan12d 6136 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)

Theoremoveqan12rd 6137 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)

Theoremoveq123d 6138 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)

Theoremnfovd 6139 Deduction version of bound-variable hypothesis builder nfov 6140. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremnfov 6140 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)

Theoremoprabid 6141 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 20-Mar-2013.)

Theoremovex 6142 The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)

Theoremovssunirn 6143 The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)

Theoremovprc 6144 The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremovprc1 6145 The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)

Theoremovprc2 6146 The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremovrcl 6147 Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)

Theoremcsbovg 6148 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremcsbov12g 6149* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Theoremcsbov1g 6150* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Theoremcsbov2g 6151* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)

Theoremrspceov 6152* A frequently used special case of rspc2ev 3069 for operation values. (Contributed by NM, 21-Mar-2007.)

Theoremfnotovb 6153 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5804. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremopabbrex 6154* A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)

Theorem0neqopab 6155 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)

Theorembrabv 6156 If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)

Theoremdfoprab2 6157* Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theoremreloprab 6158* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)

Theoremnfoprab1 6159 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremnfoprab2 6160 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)

Theoremnfoprab3 6161 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)

Theoremnfoprab 6162* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)

Theoremoprabbid 6163* Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremoprabbidv 6164* Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.)

Theoremoprabbii 6165* Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremssoprab2 6166 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4515. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremssoprab2b 6167 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4516. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Theoremeqoprab2b 6168 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4519. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremmpt2eq123 6169* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)

Theoremmpt2eq12 6170* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremmpt2eq123dva 6171* An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Theoremmpt2eq123dv 6172* An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.)

Theoremmpt2eq123i 6173 An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)

Theoremmpt2eq3dva 6174* Slightly more general equality inference for the maps to notation. (Contributed by NM, 17-Oct-2013.)

Theoremmpt2eq3ia 6175 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Theoremnfmpt21 6176 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Theoremnfmpt22 6177 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Theoremnfmpt2 6178* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Theoremoprab4 6179* Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)

Theoremcbvoprab1 6180* Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)

Theoremcbvoprab2 6181* Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcbvoprab12 6182* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremcbvoprab12v 6183* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)

Theoremcbvoprab3 6184* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)

Theoremcbvoprab3v 6185* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremcbvmpt2x 6186* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6187 allows to be a function of . (Contributed by NM, 29-Dec-2014.)

Theoremcbvmpt2 6187* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)

Theoremcbvmpt2v 6188* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4330, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Theoremelimdelov 6189 Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 25135 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremdmoprab 6190* The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremdmoprabss 6191* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)

Theoremrnoprab 6192* The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)

Theoremrnoprab2 6193* The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)

Theoremreldmoprab 6194* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)

Theoremoprabss 6195* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)

Theoremeloprabga 6196* The law of concretion for operation class abstraction. Compare elopab 4497. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremeloprabg 6197* The law of concretion for operation class abstraction. Compare elopab 4497. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremssoprab2i 6198* Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremmpt2v 6199* Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)

Theoremmpt2mptx 6200* Express a two-argument function as a one-argument function, or vice-versa. In this version is not assumed to be constant w.r.t . (Contributed by Mario Carneiro, 29-Dec-2014.)

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 32801-32900 330 32901-32936
 Copyright terms: Public domain < Previous  Next >