Home Metamath Proof ExplorerTheorem List (p. 140 of 327) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22411) Hilbert Space Explorer (22412-23934) Users' Mathboxes (23935-32663)

Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcatass 13901 Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theorem0catg 13902 Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theorem0cat 13903 The empty set is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremproplem2 13904* Lemma for mndpropd 14711. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem 13905* Lemma for mndpropd 14711. (Contributed by Mario Carneiro, 6-Dec-2014.)

Theoremproplem3 13906 Lemma for property theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)

Theoremhomffval 13907* Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfval 13908 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomffn 13909 The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
f

Theoremhomfeq 13910* Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
f f

Theoremhomfeqd 13911 If two structures have the same slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqbas 13912 Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremhomfeqval 13913 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
f f

Theoremcomfffval 13914* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomffval 13915* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomfval 13916 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf                     comp

Theoremcomfffval2 13917* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomffval2 13918* Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomfval2 13919 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf              f        comp

Theoremcomfffn 13920 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf

Theoremcomffn 13921 The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
compf

Theoremcomfeq 13922* Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp                            f f        compf compf

Theoremcomfeqd 13923 Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp comp       f f        compf compf

Theoremcomfeqval 13924 Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp       f f        compf compf

Theoremcatpropd 13925 Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)
f f        compf compf

Theoremcidpropd 13926 Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
f f        compf compf

8.1.2  Opposite category

Syntaxcoppc 13927 The opposite category operation.
oppCat

Definitiondf-oppc 13928* Define an opposite category, which is the same as the original category but with the direction of arrows the other way around. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat sSet tpos sSet comp tpos comp

Theoremoppcval 13929* Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat       sSet tpos sSet comp tpos

Theoremoppchomfval 13930 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat       tpos

Theoremoppchom 13931 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccofval 13932 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat                            comp tpos

Theoremoppcco 13933 Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       oppCat                            comp

Theoremoppcbas 13934 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccatid 13935 Lemma for oppccat 13938. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat

Theoremoppchomf 13936 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       f        tpos f

Theoremoppcid 13937 Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theoremoppccat 13938 An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
oppCat

Theorem2oppcbas 13939 The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd 13953. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              oppCat

Theorem2oppchomf 13940 The double opposite category has the same morphisms as the original category. Intended for use with property lemmas such as monpropd 13953. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat       f f oppCat

Theorem2oppccomf 13941 The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 13953. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat       compf compfoppCat

Theoremoppchomfpropd 13942 If two categories have the same hom-sets, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        f oppCat f oppCat

Theoremoppccomfpropd 13943 If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf       compfoppCat compfoppCat

8.1.3  Monomorphisms and epimorphisms

Syntaxcmon 13944 Extend class notation with the class of all monomorphisms.
Mono

Syntaxcepi 13945 Extend class notation with the class of all epimorphisms.
Epi

Definitiondf-mon 13946* Function returning the monomorphisms of the category . JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
Mono comp

Definitiondf-epi 13947 Function returning the epimorphisms of the category . JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Epi tpos MonooppCat

Theoremmonfval 13948* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp       Mono

Theoremismon 13949* Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremismon2 13950* Write out the monomorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmonhom 13951 A monomorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmoni 13952 Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Mono

Theoremmonpropd 13953 If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
f f        compf compf                     Mono Mono

Theoremoppcmon 13954 A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              Mono       Epi

Theoremoppcepi 13955 An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat              Epi       Mono

Theoremisepi 13956* Definition of an epimorphism in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremisepi2 13957* Write out the epimorphism property directly. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremepihom 13958 An epimorphism is a morphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Epi

Theoremepii 13959 Property of an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp       Epi

8.1.4  Sections, inverses, isomorphisms

Syntaxcsect 13960 Extend class notation with the sections of a morphism.
Sect

Syntaxcinv 13961 Extend class notation with the inverses of a morphism.
Inv

Syntaxciso 13962 Extend class notation with the class of all isomorphisms.

Definitiondf-sect 13963* Function returning the section relation in a category. Given arrows and , we say Sect, that is, is a section of , if . (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect comp

Definitiondf-inv 13964* The inverse relation in a category. Given arrows and , we say Inv, that is, is an inverse of , if is a section of and is a section of . (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
Inv Sect Sect

Definitiondf-iso 13965* Function returning the isomorphisms of the category . The Joy of Cats p. 28. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 2-Jan-2017.)
Inv

Theoremsectffval 13966* Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectfval 13967* Value of the section relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectss 13968 The section relation is a relation between morphisms from to and morphisms from to . (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremissect 13969 The property " is a section of ". (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremissect2 13970 Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp              Sect

Theoremsectcan 13971 If is a section of and is a section of , then . Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect

Theoremsectco 13972 Composition of two sections. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       Sect

Theoreminvffval 13973* Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoreminvfval 13974 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoremisinv 13975 Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                            Sect

Theoreminvss 13976 The inverse relation is a relation between morphisms and their inverses . (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvsym 13977 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvsym2 13978 The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvfun 13979 The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoremisoval 13980 The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminviso1 13981 If is an inverse to , then is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoreminviso2 13982 If is an inverse to , then is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoreminvf 13983 The inverse relation is a function from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvf1o 13984 The inverse relation is a bijection from isomorphisms to isomorphisms. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvinv 13985 The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv

Theoreminvco 13986 The composition of two isomorphisms is an isomorphism, and the inverse is the composition of the individual inverses. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
Inv                                          comp

Theoremisohom 13987 An isomorphism is a homomorphism. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremisoco 13988 The composition of two isomorphisms is an isomorphism. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp

Theoremoppcsect 13989 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Sect       Sect

Theoremoppcsect2 13990 A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Sect       Sect

Theoremoppcinv 13991 An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Inv       Inv

Theoremoppciso 13992 An isomorphism in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat

Theoremsectmon 13993 If is a section of , then is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono       Sect

Theoremmonsect 13994 If is a monomorphism and is a section of , then is an inverse of and they are both isomorphisms. This is also stated as "a monomorphism which is also a split epimorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono       Sect                            Inv

Theoremsectepi 13995 If is a section of , then is an epimorphism. An epimorphism that arises from a section is also known as a split epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi       Sect

Theoremepisect 13996 If is an epimorphism and is a section of , then is an inverse of and they are both isomorphisms. This is also stated as "an epimorphism which is also a split monomorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi       Sect                            Inv

8.1.5  Subcategories

Syntaxcssc 13997 Extend class notation to include the subset relation for subcategories.
cat

Syntaxcresc 13998 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
cat

Syntaxcsubc 13999 Extend class notation to include the collection of subcategories of a category.
Subcat

Definitiondf-ssc 14000* Define the subset relation for subcategories. Despite the name, this is not really a "category-aware" definition, which is to say it makes no explicit references to homsets or composition; instead this is a subset-like relation on the functions that are used as subcategory specifications in df-subc 14002, which makes it play an analogous role to the subset relation applied to the subgroups of a group. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

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 32601-32663
 Copyright terms: Public domain < Previous  Next >