Home Intuitionistic Logic ExplorerTheorem List (p. 51 of 102) < 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 - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfnbr 5001 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)

Theoremfnop 5002 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)

Theoremfneu 5003* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfneu2 5004* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)

Theoremfnun 5005 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Theoremfnunsn 5006 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfnco 5007 Composition of two functions. (Contributed by NM, 22-May-2006.)

Theoremfnresdm 5008 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)

Theoremfnresdisj 5009 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)

Theorem2elresin 5010 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)

Theoremfnssresb 5011 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)

Theoremfnssres 5012 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)

Theoremfnresin1 5013 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)

Theoremfnresin2 5014 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)

Theoremfnres 5015* An equivalence for functionality of a restriction. Compare dffun8 4929. (Contributed by Mario Carneiro, 20-May-2015.)

Theoremfnresi 5016 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)

Theoremfnima 5017 The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfn0 5018 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfnimadisj 5019 A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)

Theoremfnimaeq0 5020 Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremdfmpt3 5021 Alternate definition for the "maps to" notation df-mpt 3820. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremfnopabg 5022* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Theoremfnopab 5023* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)

Theoremmptfng 5024* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)

Theoremfnmpt 5025* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)

Theoremmpt0 5026 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremfnmpti 5027* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdmmpti 5028* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptun 5029 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremfeq1 5030 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq2 5031 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq3 5032 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq23 5033 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfeq1d 5034 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)

Theoremfeq2d 5035 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq12d 5036 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq123d 5037 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq123 5038 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)

Theoremfeq1i 5039 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq2i 5040 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)

Theoremfeq23i 5041 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq23d 5042 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)

Theoremnff 5043 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremsbcfng 5044* Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Theoremsbcfg 5045* Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Theoremffn 5046 A mapping is a function. (Contributed by NM, 2-Aug-1994.)

Theoremdffn2 5047 Any function is a mapping into . (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremffun 5048 A mapping is a function. (Contributed by NM, 3-Aug-1994.)

Theoremfrel 5049 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)

Theoremfdm 5050 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)

Theoremfdmi 5051 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)

Theoremfrn 5052 The range of a mapping. (Contributed by NM, 3-Aug-1994.)

Theoremdffn3 5053 A function maps to its range. (Contributed by NM, 1-Sep-1999.)

Theoremfss 5054 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfssd 5055 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremfco 5056 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfco2 5057 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfssxp 5058 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfex2 5059 A function with bounded domain and range is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)

Theoremfunssxp 5060 Two ways of specifying a partial function from to . (Contributed by NM, 13-Nov-2007.)

Theoremffdm 5061 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)

Theoremopelf 5062 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfun 5063 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Theoremfun2 5064 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)

Theoremfnfco 5065 Composition of two functions. (Contributed by NM, 22-May-2006.)

Theoremfssres 5066 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)

Theoremfssres2 5067 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)

Theoremfresin 5068 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremresasplitss 5069 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 5070 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfcoi2 5071 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfeu 5072* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)

Theoremfcnvres 5073 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)

Theoremfimacnvdisj 5074 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Theoremfintm 5075* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)

Theoremfin 5076 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfabexg 5077* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremfabex 5078* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)

Theoremdmfex 5079 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 5080 The empty function. (Contributed by NM, 14-Aug-1999.)

Theoremf00 5081 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)

Theoremfconst 5082 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 5083 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)

Theoremfnconstg 5084 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)

Theoremfconst6g 5085 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfconst6 5086 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)

Theoremf1eq1 5087 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq2 5088 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq3 5089 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremnff1 5090 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)

Theoremdff12 5091* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)

Theoremf1f 5092 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)

Theoremf1fn 5093 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)

Theoremf1fun 5094 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)

Theoremf1rel 5095 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremf1dm 5096 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)

Theoremf1ss 5097 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 5098 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremf1ssres 5099 A function that is one-to-one is also one-to-one on some aubset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremf1cnvcnv 5100 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.)

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 10101-10124
 Copyright terms: Public domain < Previous  Next >