Home Metamath Proof ExplorerTheorem List (p. 57 of 309) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21328) Hilbert Space Explorer (21329-22851) Users' Mathboxes (22852-30843)

Theorem List for Metamath Proof Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfex 5601 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)

Theoremfornex 5602 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)

Theoremf1dmex 5603 If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 4028. (Contributed by NM, 4-Sep-2004.)

Theoremeufnfv 5604* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)

Theoremfunfvima 5605 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)

Theoremfunfvima2 5606 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)

Theoremfunfvima3 5607 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)

Theoremfnfvima 5608 The function value of an operand in a set is contained in the image of that set, using the abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)

Theoremrexima 5609* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremralima 5610* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremidref 5611* TODO: This is the same as issref 4963 (which has a much longer proof). Should we replace issref 4963 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

Theoremfvclss 5612* Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)

Theoremfvclex 5613* Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)

Theoremfvresex 5614* Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremabrexex 5615* Existence of a class abstraction of existentially restricted sets. is normally a free-variable parameter in the class expression substituted for , which can be thought of as . This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5597, funex 5595, fnex 5593, resfunexg 5589, and funimaexg 5186. See also abrexex2 5632. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremabrexexg 5616* Existence of a class abstraction of existentially restricted sets. is normally a free-variable parameter in . The antecedent assures us that is a set. (Contributed by NM, 3-Nov-2003.)

Theoremelabrex 5617* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremabrexco 5618* Composition of two image maps and . (Contributed by NM, 27-May-2013.)

Theoremiunexg 5619* The existence of an indexed union. is normally a free-variable parameter in . (Contributed by NM, 23-Mar-2006.)

Theoremabrexex2g 5620* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremopabex3 5621* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiunex 5622* The existence of an indexed union. is normally a free-variable parameter in the class expression substituted for , which can be read informally as . (Contributed by NM, 13-Oct-2003.)

Theoremimaiun 5623* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremimauni 5624* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theoremfniunfv 5625* The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)

Theoremfuniunfv 5626* The indexed union of a function's values is the union of its image under the index class.

Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to , the theorem can be proved without this dependency. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfuniunfvf 5627* The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5626 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.)

Theoremeluniima 5628* Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)

Theoremelunirn 5629* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)

Theoremfnunirn 5630* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)

TheoremelunirnALT 5631* Membership in the union of the range of a function, proved directly. Unlike elunirn 5629, it doesn't appeal to ndmfv 5405 (via funiunfv 5626). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.)

Theoremabrexex2 5632* Existence of an existentially restricted class abstraction. is normally has free-variable parameters and . See also abrexex 5615. (Contributed by NM, 12-Sep-2004.)

Theoremabexssex 5633* Existence of a class abstraction with an existentially quantified expression. Both and can be free in . (Contributed by NM, 29-Jul-2006.)

Theoremabexex 5634* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)

Theoremdff13 5635* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)

Theoremdff13f 5636* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)

Theoremf1mpt 5637* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremf1fveq 5638 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)

Theoremf1elima 5639 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremf1imass 5640 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imaeq 5641 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremf1imapss 5642 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Theoremdff1o6 5643* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Theoremf1ocnvfv1 5644 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv2 5645 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvfv 5646 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremf1ocnvfvb 5647 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Theoremf1ocnvdm 5648 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)

Theoremfcof1 5649 An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremfcofo 5650 An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremcbvfo 5651* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremcbvexfo 5652* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)

Theoremcocan1 5653 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)

Theoremcocan2 5654 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Theoremfcof1o 5655 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremfoeqcnvco 5656 Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1eqcocnv 5657 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremfveqf1o 5658 Given a bijection , produce another bijection which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremfliftrel 5659* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel 5660* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftel1 5661* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftcnv 5662* Converse of the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfun 5663* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfund 5664* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftfuns 5665* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftf 5666* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremfliftval 5667* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremisoeq1 5668 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq2 5669 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq3 5670 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq4 5671 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremisoeq5 5672 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Theoremnfiso 5673 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremisof1o 5674 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)

Theoremisorel 5675 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)

Theoremsoisores 5676* Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsoisoi 5677* Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Theoremisoid 5678 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv 5679 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)

Theoremisocnv2 5680 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)

Theoremisocnv3 5681 Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremisores2 5682 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores1 5683 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Theoremisores3 5684 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Theoremisotr 5685 Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremisomin 5686 Isomorphisms preserve minimal elements. Note that is Takeuti and Zaring's idiom for the initial segment . Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)

Theoremisoini 5687 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)

Theoremisoini2 5688 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremisofrlem 5689* Lemma for isofr 5691. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremisoselem 5690* Lemma for isose 5692. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisofr 5691 An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremisose 5692 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
Se Se

Theoremisofr2 5693 A weak form of isofr 5691 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremisopolem 5694 Lemma for isopo 5695. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisopo 5695 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisosolem 5696 Lemma for isoso 5697. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisoso 5697 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremisowe 5698 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 5699* A weak form of isowe 5698 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theoremf1oiso 5700* 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.)

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 30801-30843
 Copyright terms: Public domain < Previous  Next >