Home Intuitionistic Logic ExplorerTheorem List (p. 59 of 105) < 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 - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremop2nd 5801 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop1std 5802 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop2ndd 5803 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop1stg 5804 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremop2ndg 5805 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremot1stg 5806 Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 5806, ot2ndg 5807, ot3rdgg 5808.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot2ndg 5807 Extract the second member of an ordered triple. (See ot1stg 5806 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot3rdgg 5808 Extract the third member of an ordered triple. (See ot1stg 5806 comment.) (Contributed by NM, 3-Apr-2015.)

Theorem1stval2 5809 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Theorem2ndval2 5810 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Theoremfo1st 5811 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo2nd 5812 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf1stres 5813 Mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf2ndres 5814 Mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo1stresm 5815* Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Theoremfo2ndresm 5816* Onto mapping of a restriction of the (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Theorem1stcof 5817 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)

Theorem2ndcof 5818 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)

Theoremxp1st 5819 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremxp2nd 5820 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorem1stexg 5821 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)

Theorem2ndexg 5822 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)

Theoremelxp6 5823 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4835. (Contributed by NM, 9-Oct-2004.)

Theoremelxp7 5824 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4835. (Contributed by NM, 19-Aug-2006.)

Theoremeqopi 5825 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)

Theoremxp2 5826* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)

Theoremunielxp 5827 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)

Theorem1st2nd2 5828 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)

Theoremxpopth 5829 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)

Theoremeqop 5830 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Theoremeqop2 5831 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)

Theoremop1steq 5832* Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)

Theorem2nd1st 5833 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)

Theorem1st2nd 5834 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)

Theorem1stdm 5835 The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)

Theorem2ndrn 5836 The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)

Theorem1st2ndbr 5837 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremreleldm2 5838* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremreldm 5839* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremsbcopeq1a 5840 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2795 that avoids the existential quantifiers of copsexg 4008). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcsbopeq1a 5841 Equality theorem for substitution of a class for an ordered pair in (analog of csbeq1a 2887). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfopab2 5842* A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab3s 5843* A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab3 5844* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

Theoremdfoprab4 5845* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab4f 5846* Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfxp3 5847* Define the cross product of three classes. Compare df-xp 4378. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)

Theoremelopabi 5848* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)

Theoremeloprabi 5849* A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremmpt2mptsx 5850* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremmpt2mpts 5851* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)

Theoremdmmpt2ssx 5852* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremfmpt2x 5853* Functionality, domain and codomain of a class given by the "maps to" notation, where is not constant but depends on . (Contributed by NM, 29-Dec-2014.)

Theoremfmpt2 5854* Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremfnmpt2 5855* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremmpt2fvex 5856* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremfnmpt2i 5857* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremdmmpt2 5858* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremmpt2fvexi 5859* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremmpt2exxg 5860* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremmpt2exg 5861* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremmpt2exga 5862* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 12-Sep-2011.)

Theoremmpt2ex 5863* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)

Theoremfmpt2co 5864* Composition of two functions. Variation of fmptco 5357 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremoprabco 5865* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Theoremoprab2co 5866* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)

Theoremdf1st2 5867* An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdf2nd2 5868* An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theorem1stconst 5869 The mapping of a restriction of the function to a constant function. (Contributed by NM, 14-Dec-2008.)

Theorem2ndconst 5870 The mapping of a restriction of the function to a converse constant function. (Contributed by NM, 27-Mar-2008.)

Theoremdfmpt2 5871* Alternate definition for the "maps to" notation df-mpt2 5544 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcnvf1olem 5872 Lemma for cnvf1o 5873. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremcnvf1o 5873* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)

Theoremf2ndf 5874 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 5875 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 5876 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 5877 Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremalgrflemg 5878 Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)

Theoremxporderlem 5879* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)

Theorempoxp 5880* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Theoremspc2ed 5881* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)

Theoremcnvoprab 5882* The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)

Theoremf1od2 5883* Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)

2.6.15  Special "Maps to" operations

The following theorems are about maps-to operations (see df-mpt2 5544) where 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 5609, ovmpt2x 5656 and fmpt2x 5853). 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.

Theoremmpt2xopn0yelv 5884* 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.)

Theoremmpt2xopoveq 5885* 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 5886* 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.)

Theoremsprmpt2 5887* 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.)

Theoremisprmpt2 5888* Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)

2.6.16  Function transposition

Syntaxctpos 5889 The transposition of a function.
tpos

Definitiondf-tpos 5890* Define the transposition of a function, which is a function tpos satisfying . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremtposss 5891 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeq 5892 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos tpos

Theoremtposeqd 5893 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
tpos tpos

Theoremtposssxp 5894 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
tpos

Theoremreltpos 5895 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos2 5896 Value of the transposition at a pair . (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtpos0 5897 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). (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremreldmtpos 5898 Necessary and sufficient condition for tpos to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
tpos

Theorembrtposg 5899 The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
tpos

Theoremottposg 5900 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
tpos

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 10401-10483
 Copyright terms: Public domain < Previous  Next >