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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfssxp 5001 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F : -->  F  C_  X.
 
Theoremfex2 5002 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.)
 F :
 -->  V  W  F  _V
 
Theoremfunssxp 5003 Two ways of specifying a partial function from to . (Contributed by NM, 13-Nov-2007.)
 Fun  F  F  C_  X.  F : dom  F -->  dom 
 F  C_
 
Theoremffdm 5004 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
 F : -->  F : dom  F -->  dom  F 
 C_
 
Theoremopelf 5005 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.)
 F :
 -->  <. C ,  D >.  F  C  D
 
Theoremfun 5006 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
 F : --> C  G : --> D  i^i  (/)  F  u.  G :  u.  --> C  u.  D
 
Theoremfun2 5007 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
 F : --> C  G : --> C  i^i  (/)  F  u.  G :  u.  --> C
 
Theoremfnfco 5008 Composition of two functions. (Contributed by NM, 22-May-2006.)
 F  Fn  G : -->  F  o.  G  Fn
 
Theoremfssres 5009 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
 F :
 -->  C  C_  F  |`  C : C -->
 
Theoremfssres2 5010 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
 F  |`  : -->  C  C_  F  |`  C : C -->
 
Theoremfresin 5011 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 F : -->  F  |`  X :  i^i  X -->
 
Theoremresasplitss 5012 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.)
 F  Fn  G  Fn  F  |` 
 i^i  G  |` 
 i^i  F  |` 
 i^i  u.  F  |` 
 \  u.  G  |` 
 \  C_  F  u.  G
 
Theoremfcoi1 5013 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F : -->  F  o.  _I  |`  F
 
Theoremfcoi2 5014 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F : -->  _I  |`  o.  F  F
 
Theoremfeu 5015* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
 F :
 -->  C  <. C ,  >.  F
 
Theoremfcnvres 5016 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
 F : -->  `' F  |`  `' F  |`
 
Theoremfimacnvdisj 5017 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
 F :
 --> 
 i^i  C  (/)  `' F " C  (/)
 
Theoremfintm 5018* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
   =>     F : --> |^|  F : -->
 
Theoremfin 5019 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F : -->  i^i  C  F : -->  F : --> C
 
Theoremfabexg 5020* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
 F  {  |  : -->  }   =>     C  D  F  _V
 
Theoremfabex 5021* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
 _V   &     _V   &     F  {  |  : -->  }   =>     F  _V
 
Theoremdmfex 5022 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.)
 F  C  F : -->  _V
 
Theoremf0 5023 The empty function. (Contributed by NM, 14-Aug-1999.)
 (/) : (/) -->
 
Theoremf00 5024 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
 F : --> (/)  F  (/)  (/)
 
Theoremfconst 5025 A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 _V   =>     X.  { } : --> { }
 
Theoremfconstg 5026 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
 V  X.  { } : --> { }
 
Theoremfnconstg 5027 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
 V  X.  { }  Fn
 
Theoremfconst6g 5028 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 C  X.  { } : --> C
 
Theoremfconst6 5029 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
 C   =>     X.  { } : --> C
 
Theoremf1eq1 5030 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
 F  G  F : -1-1->  G : -1-1->
 
Theoremf1eq2 5031 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
 F : -1-1-> C  F : -1-1-> C
 
Theoremf1eq3 5032 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
 F : C -1-1->  F : C -1-1->
 
Theoremnff1 5033 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
 F/_ F   &     F/_   &     F/_   =>    
 F/  F :
 -1-1->
 
Theoremdff12 5034* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
 F : -1-1->  F : -->  F
 
Theoremf1f 5035 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
 F : -1-1->  F : -->
 
Theoremf1fn 5036 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
 F : -1-1->  F  Fn
 
Theoremf1fun 5037 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
 F : -1-1->  Fun  F
 
Theoremf1rel 5038 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
 F : -1-1->  Rel  F
 
Theoremf1dm 5039 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
 F : -1-1->  dom  F
 
Theoremf1ss 5040 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.)
 F :
 -1-1->  C_  C  F :
 -1-1-> C
 
Theoremf1ssr 5041 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 F :
 -1-1->  ran  F  C_  C  F : -1-1-> C
 
Theoremf1ssres 5042 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.)
 F :
 -1-1->  C  C_  F  |`  C : C -1-1->
 
Theoremf1cnvcnv 5043 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.)
 `' `' : dom  -1-1-> _V  Fun  `'  Fun  `' `'
 
Theoremf1co 5044 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)
 F :
 -1-1-> C  G :
 -1-1->  F  o.  G :
 -1-1-> C
 
Theoremfoeq1 5045 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 F  G  F : -onto->  G : -onto->
 
Theoremfoeq2 5046 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 F : -onto-> C  F : -onto-> C
 
Theoremfoeq3 5047 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 F : C -onto->  F : C -onto->
 
Theoremnffo 5048 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
 F/_ F   &     F/_   &     F/_   =>    
 F/  F : -onto->
 
Theoremfof 5049 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
 F : -onto->  F : -->
 
Theoremfofun 5050 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
 F : -onto->  Fun  F
 
Theoremfofn 5051 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
 F : -onto->  F  Fn
 
Theoremforn 5052 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
 F : -onto->  ran  F
 
Theoremdffo2 5053 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
 F : -onto->  F : -->  ran  F
 
Theoremfoima 5054 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
 F : -onto->  F "
 
Theoremdffn4 5055 A function maps onto its range. (Contributed by NM, 10-May-1998.)
 F  Fn  F : -onto-> ran  F
 
Theoremfunforn 5056 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
 Fun  : dom  -onto-> ran
 
Theoremfodmrnu 5057 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
 F : -onto->  F : C -onto-> D  C  D
 
Theoremfores 5058 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
 Fun  F  C_  dom  F  F  |`  : -onto-> F
 "
 
Theoremfoco 5059 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
 F : -onto-> C  G : -onto->  F  o.  G : -onto-> C
 
Theoremf1oeq1 5060 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 F  G  F : -1-1-onto->  G : -1-1-onto->
 
Theoremf1oeq2 5061 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 F : -1-1-onto-> C  F : -1-1-onto-> C
 
Theoremf1oeq3 5062 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 F : C -1-1-onto->  F : C -1-1-onto->
 
Theoremf1oeq23 5063 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
 C  D  F :
 -1-1-onto-> C  F : -1-1-onto-> D
 
Theoremf1eq123d 5064 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 F  G   &       &     C  D   =>     F : -1-1-> C  G :
 -1-1-> D
 
Theoremfoeq123d 5065 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 F  G   &       &     C  D   =>     F : -onto-> C  G : -onto-> D
 
Theoremf1oeq123d 5066 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 F  G   &       &     C  D   =>     F : -1-1-onto-> C  G : -1-1-onto-> D
 
Theoremnff1o 5067 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
 F/_ F   &     F/_   &     F/_   =>    
 F/  F :
 -1-1-onto->
 
Theoremf1of1 5068 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
 F : -1-1-onto->  F : -1-1->
 
Theoremf1of 5069 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
 F : -1-1-onto->  F : -->
 
Theoremf1ofn 5070 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
 F : -1-1-onto->  F  Fn
 
Theoremf1ofun 5071 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
 F : -1-1-onto->  Fun  F
 
Theoremf1orel 5072 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
 F : -1-1-onto->  Rel  F
 
Theoremf1odm 5073 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
 F : -1-1-onto->  dom  F
 
Theoremdff1o2 5074 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 F : -1-1-onto->  F  Fn 
 Fun  `' F  ran  F
 
Theoremdff1o3 5075 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 F : -1-1-onto->  F : -onto-> 
 Fun  `' F
 
Theoremf1ofo 5076 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
 F : -1-1-onto->  F : -onto->
 
Theoremdff1o4 5077 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 F : -1-1-onto->  F  Fn  `' F  Fn
 
Theoremdff1o5 5078 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 F : -1-1-onto->  F : -1-1-> 
 ran  F
 
Theoremf1orn 5079 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
 F : -1-1-onto-> ran  F  F  Fn 
 Fun  `' F
 
Theoremf1f1orn 5080 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
 F : -1-1->  F : -1-1-onto-> ran  F
 
Theoremf1oabexg 5081* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
 F  {  |  : -1-1-onto->  }   =>     C  D  F  _V
 
Theoremf1ocnv 5082 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.)
 F : -1-1-onto->  `' F : -1-1-onto->
 
Theoremf1ocnvb 5083 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.)
 Rel  F  F : -1-1-onto->  `' F :
 -1-1-onto->
 
Theoremf1ores 5084 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
 F :
 -1-1->  C  C_  F  |`  C : C -1-1-onto-> F " C
 
Theoremf1orescnv 5085 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
 Fun  `' F  F  |`  R : R -1-1-onto-> P  `' F  |`  P : P -1-1-onto-> R
 
Theoremf1imacnv 5086 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
 F :
 -1-1->  C  C_  `' F " F " C  C
 
Theoremfoimacnv 5087 A reverse version of f1imacnv 5086. (Contributed by Jeff Hankins, 16-Jul-2009.)
 F : -onto->  C  C_  F
 " `' F " C  C
 
Theoremfoun 5088 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
 F : -onto->  G : C -onto-> D  i^i  C  (/)  F  u.  G :  u.  C -onto->  u.  D
 
Theoremf1oun 5089 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
 F : -1-1-onto->  G : C
 -1-1-onto-> D  i^i  C  (/)  i^i  D  (/)  F  u.  G :  u.  C
 -1-1-onto->  u.  D
 
Theoremfun11iun 5090* 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.)
 C   &     _V   =>     : D -1-1-> S  C_  C  C  C_  U_  : U_  D -1-1-> S
 
Theoremresdif 5091 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.)
 Fun  `' F  F  |`  : -onto-> C  F  |`  : -onto-> D  F  |` 
 \  :  \  -1-1-onto-> C  \  D
 
Theoremf1oco 5092 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
 F :
 -1-1-onto-> C  G : -1-1-onto->  F  o.  G : -1-1-onto-> C
 
Theoremf1cnv 5093 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
 F : -1-1->  `' F : ran  F -1-1-onto->
 
Theoremfuncocnv2 5094 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
 Fun  F  F  o.  `' F  _I  |` 
 ran  F
 
Theoremfococnv2 5095 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 F : -onto->  F  o.  `' F  _I  |`
 
Theoremf1ococnv2 5096 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.)
 F : -1-1-onto->  F  o.  `' F  _I  |`
 
Theoremf1cocnv2 5097 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
 F : -1-1->  F  o.  `' F  _I  |`  ran  F
 
Theoremf1ococnv1 5098 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.)
 F : -1-1-onto->  `' F  o.  F  _I  |`
 
Theoremf1cocnv1 5099 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
 F : -1-1->  `' F  o.  F  _I  |`
 
Theoremfuncoeqres 5100 Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 Fun  G  F  o.  G  H  F  |`  ran  G  H  o.  `' G
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