Home Metamath Proof ExplorerTheorem List (p. 58 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfvex 5701 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)

Theoremfvif 5702 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfv3 5703* Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvres 5704 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)

Theoremfunssfv 5705 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)

Theoremtz6.12-1 5706* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremtz6.12 5707* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)

Theoremtz6.12f 5708* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)

Theoremtz6.12c 5709* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremtz6.12i 5710 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremfvbr0 5711 Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfvrn0 5712 A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)

Theoremfvssunirn 5713 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremndmfv 5714 The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremndmfvrcl 5715 Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)

Theoremelfvdm 5716 If a function value has a member, the argument belongs to the domain. (Contributed by NM, 12-Feb-2007.)

Theoremelfvex 5717 If a function value has a member, the argument is a set. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremelfvexd 5718 If a function value is nonempty, its argument is a set. Deduction form of elfvex 5717. (Contributed by David Moews, 1-May-2017.)

Theoremnfvres 5719 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)

Theoremnfunsn 5720 If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfvfundmfvn0 5721 If a class' value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)

Theoremfv01 5722 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Theoremfveqres 5723 Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)

Theoremfunbrfv 5724 The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfunopfv 5725 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)

Theoremfnbrfvb 5726 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfnopfvb 5727 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)

Theoremfunbrfvb 5728 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)

Theoremfunopfvb 5729 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)

Theoremfunbrfv2b 5730 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremdffn5 5731* Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfnrnfv 5732* The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfvelrnb 5733* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Theoremdfimafn 5734* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremdfimafn2 5735* Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass4 5736* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfvelima 5737* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfeqmptd 5738* Deduction form of dffn5 5731. (Contributed by Mario Carneiro, 8-Jan-2015.)

Theoremfeqresmpt 5739* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdffn5f 5740* Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)

Theoremfvelimab 5741* Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)

Theoremfvi 5742 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfviss 5743 The value of the identity function is a subset of the argument. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theoremfniinfv 5744* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)

Theoremfnsnfv 5745 Singleton of function value. (Contributed by NM, 22-May-1998.)

Theoremfnimapr 5746 The image of a pair under a funtion. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremssimaex 5747* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)

Theoremssimaexg 5748* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Theoremfunfv 5749 A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)

Theoremfunfv2 5750* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)

Theoremfunfv2f 5751 The value of a function. Version of funfv2 5750 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)

Theoremfvun 5752 Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)

Theoremfvun1 5753 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremfvun2 5754 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremdffv2 5755 Alternate definition of function value df-fv 5421 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)

Theoremdmfco 5756 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)

Theoremfvco2 5757 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)

Theoremfvco 5758 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)

Theoremfvco3 5759 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremfvco4i 5760 Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)

Theoremfvopab3g 5761* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfvopab3ig 5762* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremfvmptg 5763* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpti 5764* Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfvmpt 5765* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)

Theoremfvmpts 5766* Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt3 5767* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremfvmpt3i 5768* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptd 5769* Deduction version of fvmpt 5765. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt2i 5770* Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremfvmpt2 5771* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)

Theoremfvmptss 5772* If all the values of the mapping are subsets of a class , then so is any evaluation of the mapping, even if is not in the base set . (Contributed by Mario Carneiro, 13-Feb-2015.)

Theoremfvmpt2d 5773* Deduction version of fvmpt2 5771. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Theoremfvmptex 5774* Express a function whose value may not always be a set in terms of another function for which sethood is guaranteed. (Note that is just shorthand for , and it is always a set by fvex 5701.) Note also that these functions are not the same; wherever is not a set, is not in the domain of (so it evaluates to the empty set), but is in the domain of , and is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremfvmptdf 5775* Alternate deduction version of fvmpt 5765, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv 5776* Alternate deduction version of fvmpt 5765, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv2 5777* Alternate deduction version of fvmpt 5765, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremmpteqb 5778* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5786. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremfvmptt 5779* Closed theorem form of fvmpt 5765. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptf 5780* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5763 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfvmptnf 5781* The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 5782 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptn 5782* This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 5763. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)

Theoremfvmptss2 5783* A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)

Theoremfvopab4ndm 5784* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)

Theoremfvopab6 5785* Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremeqfnfv 5786* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremeqfnfv2 5787* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremeqfnfv3 5788* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeqfnfvd 5789* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremeqfnfv2f 5790* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5786 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Theoremeqfunfv 5791* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Theoremfvreseq 5792* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)

Theoremfndmdif 5793* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifcom 5794 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifeq0 5795 The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmin 5796* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfneqeql 5797 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremfneqeql2 5798 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremfnreseql 5799 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremchfnrn 5800* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)

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 32401-32447
 Copyright terms: Public domain < Previous  Next >