Home Intuitionistic Logic ExplorerTheorem List (p. 54 of 133) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfresin 5301 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremresasplitss 5302 If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)

Theoremfcoi1 5303 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfcoi2 5304 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfeu 5305* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)

Theoremfcnvres 5306 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)

Theoremfimacnvdisj 5307 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Theoremfintm 5308* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)

Theoremfin 5309 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfabexg 5310* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremfabex 5311* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)

Theoremdmfex 5312 If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremf0 5313 The empty function. (Contributed by NM, 14-Aug-1999.)

Theoremf00 5314 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)

Theoremf0bi 5315 A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)

Theoremf0dom0 5316 A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)

Theoremf0rn0 5317* If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)

Theoremfconst 5318 A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfconstg 5319 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)

Theoremfnconstg 5320 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)

Theoremfconst6g 5321 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfconst6 5322 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)

Theoremf1eq1 5323 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq2 5324 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq3 5325 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremnff1 5326 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)

Theoremdff12 5327* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)

Theoremf1f 5328 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)

Theoremf1rn 5329 The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)

Theoremf1fn 5330 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)

Theoremf1fun 5331 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)

Theoremf1rel 5332 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremf1dm 5333 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)

Theoremf1ss 5334 A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremf1ssr 5335 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremf1ff1 5336 If a function is one-to-one from A to B and is also a function from A to C, then it is a one-to-one function from A to C. (Contributed by BJ, 4-Jul-2022.)

Theoremf1ssres 5337 A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremf1resf1 5338 The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.)

Theoremf1cnvcnv 5339 Two ways to express that a set (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)

Theoremf1co 5340 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)

Theoremfoeq1 5341 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq2 5342 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq3 5343 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremnffo 5344 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)

Theoremfof 5345 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)

Theoremfofun 5346 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)

Theoremfofn 5347 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)

Theoremforn 5348 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)

Theoremdffo2 5349 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)

Theoremfoima 5350 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Theoremdffn4 5351 A function maps onto its range. (Contributed by NM, 10-May-1998.)

Theoremfunforn 5352 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)

Theoremfodmrnu 5353 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)

Theoremfores 5354 Restriction of a function. (Contributed by NM, 4-Mar-1997.)

Theoremfoco 5355 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)

Theoremf1oeq1 5356 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq2 5357 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq3 5358 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq23 5359 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Theoremf1eq123d 5360 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremfoeq123d 5361 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremf1oeq123d 5362 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremf1oeq2d 5363 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Theoremf1oeq3d 5364 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Theoremnff1o 5365 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)

Theoremf1of1 5366 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1of 5367 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofn 5368 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofun 5369 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)

Theoremf1orel 5370 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)

Theoremf1odm 5371 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)

Theoremdff1o2 5372 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o3 5373 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ofo 5374 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)

Theoremdff1o4 5375 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o5 5376 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1orn 5377 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Theoremf1f1orn 5378 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)

Theoremf1oabexg 5379* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremf1ocnv 5380 The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ocnvb 5381 A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)

Theoremf1ores 5382 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)

Theoremf1orescnv 5383 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremf1imacnv 5384 Preimage of an image. (Contributed by NM, 30-Sep-2004.)

Theoremfoimacnv 5385 A reverse version of f1imacnv 5384. (Contributed by Jeff Hankins, 16-Jul-2009.)

Theoremfoun 5386 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremf1oun 5387 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)

Theoremfun11iun 5388* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremresdif 5389 The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremf1oco 5390 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)

Theoremf1cnv 5391 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)

Theoremfuncocnv2 5392 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfococnv2 5393 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1ococnv2 5394 The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)

Theoremf1cocnv2 5395 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremf1ococnv1 5396 The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)

Theoremf1cocnv1 5397 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremfuncoeqres 5398 Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremffoss 5399* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)

Theoremf11o 5400* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)

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 13201-13283
 Copyright terms: Public domain < Previous  Next >