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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdf1st2 6901* An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdf2nd2 6902* An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theorem1stconst 6903 The mapping of a restriction of the function to a constant function. (Contributed by NM, 14-Dec-2008.)

Theorem2ndconst 6904 The mapping of a restriction of the function to a converse constant function. (Contributed by NM, 27-Mar-2008.)

Theoremdfmpt2 6905* Alternate definition for the "maps to" notation df-mpt2 6313 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmpt2sn 6906* An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)

Theoremcurry1 6907* Composition with turns any binary operation with a constant first operand into a function of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremcurry1val 6908 The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremcurry1f 6909 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)

Theoremcurry2 6910* Composition with turns any binary operation with a constant second operand into a function of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)

Theoremcurry2f 6911 Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)

Theoremcurry2val 6912 The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)

Theoremcnvf1olem 6913 Lemma for cnvf1o 6914. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremcnvf1o 6914* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremfparlem1 6915 Lemma for fpar 6919. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem2 6916 Lemma for fpar 6919. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem3 6917* Lemma for fpar 6919. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfparlem4 6918* Lemma for fpar 6919. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfpar 6919* Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as . (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfsplit 6920 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 6919 in order to build compound functions such as . (Contributed by NM, 17-Sep-2007.)

Theoremf2ndf 6921 The (second member of an ordered pair) function restricted to a function is a function of into the codomain of . (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Theoremfo2ndf 6922 The (second member of an ordered pair) function restricted to a function is a function of onto the range of . (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Theoremf1o2ndf1 6923 The (second member of an ordered pair) function restricted to a one-to-one function is a one-to-one function of onto the range of . (Contributed by Alexander van der Vekens, 4-Feb-2018.)

Theoremalgrflem 6924 Lemma for algrf 14611 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfrxp 6925* A lexicographical ordering of two well-founded classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) (Proof shortened by Wolf Lammen, 4-Oct-2014.)

Theoremxporderlem 6926* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)

Theorempoxp 6927* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremsoxp 6928* A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremwexp 6929* A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremfnwelem 6930* Lemma for fnwe 6931. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremfnwe 6931* A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremfnse 6932* Condition for the well-order in fnwe 6931 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
Se               Se

2.4.8  The support of functions

In this section, the support of functions is defined and corresponding theorems are provided. Since basic properties (see suppval 6935) are based on the Axiom of Union (usage of dmexg 6743), these definition and theorems cannot be provided earlier. Until April 2019, the support of a function was represented by the expression (see suppimacnv 6944). The theorems which are based on this representation and which are provided in previous sections could be moved into this section to have all related theorems in one section, although they do not depend on the Axiom of Union. This was possible because they are not used before. The current theorems differ from the original ones by requiring that the classes representing the "function" (or its "domain") and the "zero element" are sets. Actually, this does not cause any problem (until now).

Syntaxcsupp 6933 Extend class definition to include the support of functions.
supp

Definitiondf-supp 6934* Define the support of a function against a "zero" value. According to Wikipedia ("Support (mathematics)", 31-Mar-2019, https://en.wikipedia.org/wiki/Support_(mathematics)) "In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero." and "The notion of support also extends in a natural way to functions taking values in more general sets than R [the real numbers] and to other objects.". The following definition allows for such extensions, being applicable for any sets (which usually are functions) and any element (even not necessarily from the range of the function) regarded as "zero". (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
supp

Theoremsuppval 6935* The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
supp

Theoremsupp0prc 6936 The support of a class is empty if either the class or the "zero" is a proper class. . (Contributed by AV, 28-May-2019.)
supp

Theoremsuppvalbr 6937* The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
supp

Theoremsupp0 6938 The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
supp

Theoremsuppval1 6939* The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
supp

Theoremsuppvalfn 6940* The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
supp

Theoremelsuppfn 6941 An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
supp

Theoremcnvimadfsn 6942* The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)

Theoremsuppimacnvss 6943 The support of functions "defined" by inverse images is a subset of the support defined by df-supp 6934. (Contributed by AV, 7-Apr-2019.)
supp

Theoremsuppimacnv 6944 Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
supp

Theoremfrnsuppeq 6945 Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
supp

Theoremsuppssdm 6946 The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.)
supp

Theoremsuppsnop 6947 The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
supp

Theoremsnopsuppss 6948 The support of a singleton containing an ordered pair is a subset of the singleton containing the first element of the ordered pair, i.e. it is empty or the singleton itself. (Contributed by AV, 19-Jul-2019.)
supp

Theoremfvn0elsupp 6949 If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)
supp

Theoremfvn0elsuppOLD 6950 Obsolete version of fvn0elsupp 6949 as of 4-Apr-2020. (Contributed by AV, 2-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
supp

Theoremfvn0elsuppb 6951 The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)
supp

Theoremrexsupp 6952* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)
supp

Theoremressuppss 6953 The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
supp supp

Theoremsuppun 6954 The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.)
supp supp

Theoremressuppssdif 6955 The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
supp supp

Theoremmptsuppdifd 6956* The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
supp

Theoremmptsuppd 6957* The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
supp

Theoremextmptsuppeq 6958* The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 30-Jun-2019.)
supp supp

Theoremsuppfnss 6959* The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.)
supp supp

Theoremfunsssuppss 6960 The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
supp supp

Theoremfnsuppres 6961 Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.)
supp

Theoremfnsuppeq0 6962 The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
supp

Theoremfczsupp0 6963 The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
supp

Theoremsuppss 6964* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
supp

Theoremsuppssr 6965 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
supp

Theoremsuppssov1 6966* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
supp                                    supp

Theoremsuppssof1 6967* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
supp                                           supp

Theoremsuppss2 6968* 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.) (Revised by AV, 28-May-2019.)
supp

Theoremsuppsssn 6969* Show that the support of a function is a subset of a singleton. (Contributed by AV, 21-Jul-2019.)
supp

Theoremsuppssfv 6970* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
supp                             supp

Theoremsuppofss1d 6971* Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
supp supp

Theoremsuppofss2d 6972* Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
supp supp

Theoremsupp0cosupp0 6973 The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
supp supp

Theoremimacosupp 6974 The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
supp supp supp

2.4.9  Special "Maps to" operations

The following theorems are about maps-to operations (see df-mpt2 6313) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 6388, ovmpt2x 6444 and fmpt2x 6878). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short.

Theoremopeliunxp2f 6975* Membership in a union of Cartesian products, using bound-variable hypothesis for instead of distinct variable conditions as in opeliunxp2 4978. (Contributed by AV, 25-Oct-2020.)

Theoremmpt2xeldm 6976* If there is an element of the value of an operation given by a maps-to rule, then the first argument is an element of the first component of the domain and the second argument is an element of the second component of the domain depending on the first argument. (Contributed by AV, 25-Oct-2020.)

Theoremmpt2xneldm 6977* If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.)

Theoremmpt2xopn0yelv 6978* If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopynvov0g 6979* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopxnop0 6980* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopx0ov0 6981* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopxprcov0 6982* If the components of the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopynvov0 6983* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Theoremmpt2xopoveq 6984* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)

Theoremmpt2xopovel 6985* Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)

Theoremmpt2xopoveqd 6986* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)

Theorembrovex 6987* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)

Theorembrovmpt2ex 6988* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)

Theoremsprmpt2d 6989* The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.)

2.4.10  Function transposition

Syntaxctpos 6990 The transposition of a function.
tpos

Definitiondf-tpos 6991* Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposss 6992 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeq 6993 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeqd 6994 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
tpos tpos

Theoremtposssxp 6995 The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos

Theoremreltpos 6996 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos2 6997 Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos0 6998 The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows us to eliminate sethood hypotheses on in brtpos 7000. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremreldmtpos 6999 Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos 7000 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

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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 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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