Home Metamath Proof ExplorerTheorem List (p. 50 of 329) < 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 (1-22452) Hilbert Space Explorer (22453-23975) Users' Mathboxes (23976-32860)

Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfindsg 4901* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. The basis of this version is an arbitrary natural number instead of zero. (Contributed by NM, 16-Sep-1995.)

Theoremfinds2 4902* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)

Theoremfinds1 4903* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)

Theoremfindes 4904 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. See tfindes 4871 for the transfinite version. (Contributed by Raph Levien, 9-Jul-2003.)

2.4.7  Relations

Syntaxcxp 4905 Extend the definition of a class to include the cross product.

Syntaxccnv 4906 Extend the definition of a class to include the converse of a class.

Syntaxcdm 4907 Extend the definition of a class to include the domain of a class.

Syntaxcrn 4908 Extend the definition of a class to include the range of a class.

Syntaxcres 4909 Extend the definition of a class to include the restriction of a class. (Read: The restriction of to .)

Syntaxcima 4910 Extend the definition of a class to include the image of a class. (Read: The image of under .)

Syntaxccom 4911 Extend the definition of a class to include the composition of two classes. (Read: The composition of and .)

Syntaxwrel 4912 Extend the definition of a wff to include the relation predicate. (Read: is a relation.)

Definitiondf-xp 4913* Define the cross product of two classes. Definition 9.11 of [Quine] p. 64. For example, (ex-xp 21775). Another example is that the set of rational numbers are defined in df-q 10606 using the cross-product ; the left- and right-hand sides of the cross-product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)

Definitiondf-rel 4914 Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5350 and dfrel3 5357. (Contributed by NM, 1-Aug-1994.)

Definitiondf-cnv 4915* Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if and then , as proven in brcnv 5084 (see df-br 4238 and df-rel 4914 for more on relations). For example, (ex-cnv 21776). We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)

Definitiondf-co 4916* Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. For example, (ex-co 21777) because (see cos0 12782) and (see df-e 12702). Note that Definition 7 of [Suppes] p. 63 reverses and , uses instead of , and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.)

Definitiondf-dm 4917* Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, (ex-dm 21778). Another example is the domain of the complex arctangent, arctan (for proof see atandm 20747). Contrast with range (defined in df-rn 4918). For alternate definitions see dfdm2 5430, dfdm3 5087, and dfdm4 5092. The notation " " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)

Definitiondf-rn 4918 Define the range of a class. For example, (ex-rn 21779). Contrast with domain (defined in df-dm 4917). For alternate definitions, see dfrn2 5088, dfrn3 5089, and dfrn4 5360. The notation " " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)

Definitiondf-res 4919 Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, the expression (used in reeff1 12752) means "the exponential function e to the x, but the exponent x must be in the reals" (df-ef 12701 defines the exponential function, which normally allows the exponent to be a complex number). Another example is that (ex-res 21780). (Contributed by NM, 2-Aug-1994.)

Definitiondf-ima 4920 Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example, (ex-ima 21781). Contrast with restriction (df-res 4919) and range (df-rn 4918). For an alternate definition, see dfima2 5234. (Contributed by NM, 2-Aug-1994.)

Theoremxpeq1 4921 Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)

Theoremxpeq2 4922 Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)

Theoremelxpi 4923* Membership in a cross product. Uses fewer axioms than elxp 4924. (Contributed by NM, 4-Jul-1994.)

Theoremelxp 4924* Membership in a cross product. (Contributed by NM, 4-Jul-1994.)

Theoremelxp2 4925* Membership in a cross product. (Contributed by NM, 23-Feb-2004.)

Theoremxpeq12 4926 Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)

Theoremxpeq1i 4927 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)

Theoremxpeq2i 4928 Equality inference for cross product. (Contributed by NM, 21-Dec-2008.)

Theoremxpeq12i 4929 Equality inference for cross product. (Contributed by FL, 31-Aug-2009.)

Theoremxpeq1d 4930 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremxpeq2d 4931 Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremxpeq12d 4932 Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.)

Theoremnfxp 4933 Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbxpg 4934 Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)

Theorem0nelxp 4935 The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorem0nelelxp 4936 A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)

Theoremopelxp 4937 Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrxp 4938 Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.)

Theoremopelxpi 4939 Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)

Theoremopelxp1 4940 The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopelxp2 4941 The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremotelxp1 4942 The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)

Theoremrabxp 4943* Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)

Theorembrrelex12 4944 A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorembrrelex 4945 A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrrelex2 4946 A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theorembrrelexi 4947 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)

Theorembrrelex2i 4948 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremnprrel 4949 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)

Theoremfconstmpt 4950* Representation of a constant function using the mapping operation. (Note that cannot appear free in .) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremvtoclr 4951* Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopelvvg 4952 Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)

Theoremopelvv 4953 Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremopthprc 4954 Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)

Theorembrel 4955 Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrab2a 4956* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)

Theoremelxp3 4957* Membership in a cross product. (Contributed by NM, 5-Mar-1995.)

Theoremopeliunxp 4958 Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)

Theoremxpundi 4959 Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)

Theoremxpundir 4960 Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)

Theoremxpiundi 4961* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremxpiundir 4962* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremiunxpconst 4963* Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremxpun 4964 The cross product of two unions. (Contributed by NM, 12-Aug-2004.)

Theoremelvv 4965* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)

Theoremelvvv 4966* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)

Theoremelvvuni 4967 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)

Theorembrinxp2 4968 Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorembrinxp 4969 Intersection of binary relation with cross product. (Contributed by NM, 9-Mar-1997.)

Theorempoinxp 4970 Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremsoinxp 4971 Intersection of total order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremfrinxp 4972 Intersection of well-founded relation with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremseinxp 4973 Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se Se

Theoremweinxp 4974 Intersection of well-ordering with cross product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)

Theoremposn 4975 Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremsosn 4976 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremfrsn 4977 Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremwesn 4978 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremopabssxp 4979* An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)

Theorembrab2ga 4980* The law of concretion for a binary relation. See brab2a 4956 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)

Theoremoptocl 4981* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)

Theorem2optocl 4982* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theorem3optocl 4983* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theoremopbrop 4984* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)

Theoremxp0r 4985 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)

Theoremonxpdisj 4986 Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 4729. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremonnev 4987 The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.)

Theoremreleq 4988 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)

Theoremreleqi 4989 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)

Theoremreleqd 4990 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)

Theoremnfrel 4991 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrelss 4992 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)

Theoremssrel 4993* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremeqrel 4994* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)

Theoremssrel2 4995* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4993 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)

Theoremrelssi 4996* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)

Theoremrelssdv 4997* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)

Theoremeqrelriv 4998* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)

Theoremeqrelriiv 4999* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)

Theoremeqbrriv 5000* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)

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 32801-32860
 Copyright terms: Public domain < Previous  Next >