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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.)
 |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )
 
Theoremfnop 5002 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
 |-  ( ( F  Fn  A  /\  <. B ,  C >.  e.  F )  ->  B  e.  A )
 
Theoremfneu 5003* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
 
Theoremfneu2 5004* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y <. B ,  y >.  e.  F )
 
Theoremfnun 5005 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
 |-  ( ( ( F  Fn  A  /\  G  Fn  B )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B ) )
 
Theoremfnunsn 5006 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  Y  e.  _V )   &    |-  ( ph  ->  F  Fn  D )   &    |-  G  =  ( F  u.  { <. X ,  Y >. } )   &    |-  E  =  ( D  u.  { X } )   &    |-  ( ph  ->  -.  X  e.  D )   =>    |-  ( ph  ->  G  Fn  E )
 
Theoremfnco 5007 Composition of two functions. (Contributed by NM, 22-May-2006.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G )  Fn  B )
 
Theoremfnresdm 5008 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
 |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
 
Theoremfnresdisj 5009 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
 |-  ( F  Fn  A  ->  ( ( A  i^i  B )  =  (/)  <->  ( F  |`  B )  =  (/) ) )
 
Theorem2elresin 5010 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
 |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G )  <->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) )
 
Theoremfnssresb 5011 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
 |-  ( F  Fn  A  ->  ( ( F  |`  B )  Fn  B  <->  B  C_  A ) )
 
Theoremfnssres 5012 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( F  |`  B )  Fn  B )
 
Theoremfnresin1 5013 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
 |-  ( F  Fn  A  ->  ( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )
 
Theoremfnresin2 5014 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
 |-  ( F  Fn  A  ->  ( F  |`  ( B  i^i  A ) )  Fn  ( B  i^i  A ) )
 
Theoremfnres 5015* An equivalence for functionality of a restriction. Compare dffun8 4929. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ( F  |`  A )  Fn  A  <->  A. x  e.  A  E! y  x F y )
 
Theoremfnresi 5016 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
 |-  (  _I  |`  A )  Fn  A
 
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.)
 |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
 
Theoremfn0 5018 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F  Fn  (/)  <->  F  =  (/) )
 
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.)
 |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( F " C )  =  (/) )
 
Theoremfnimaeq0 5020 Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( ( F
 " B )  =  (/) 
 <->  B  =  (/) ) )
 
Theoremdfmpt3 5021 Alternate definition for the "maps to" notation df-mpt 3820. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
 )
 
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.)
 |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }   =>    |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A )
 
Theoremfnopab 5023* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
 |-  ( x  e.  A  ->  E! y ph )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }   =>    |-  F  Fn  A
 
Theoremmptfng 5024* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A )
 
Theoremfnmpt 5025* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  V  ->  F  Fn  A )
 
Theoremmpt0 5026 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( x  e.  (/)  |->  A )  =  (/)
 
Theoremfnmpti 5027* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  F  Fn  A
 
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.)
 |-  B  e.  _V   &    |-  F  =  ( x  e.  A  |->  B )   =>    |- 
 dom  F  =  A
 
Theoremmptun 5029 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( x  e.  ( A  u.  B )  |->  C )  =  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )
 
Theoremfeq1 5030 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F : A --> B 
 <->  G : A --> B ) )
 
Theoremfeq2 5031 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : A --> C 
 <->  F : B --> C ) )
 
Theoremfeq3 5032 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : C --> A 
 <->  F : C --> B ) )
 
Theoremfeq23 5033 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A
 --> B  <->  F : C --> D ) )
 
Theoremfeq1d 5034 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F : A --> B  <->  G : A --> B ) )
 
Theoremfeq2d 5035 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A --> C  <->  F : B --> C ) )
 
Theoremfeq12d 5036 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A --> C  <->  G : B --> C ) )
 
Theoremfeq123d 5037 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( F : A --> C  <->  G : B --> D ) )
 
Theoremfeq123 5038 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D ) 
 ->  ( F : A --> B 
 <->  G : C --> D ) )
 
Theoremfeq1i 5039 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F : A
 --> B  <->  G : A --> B )
 
Theoremfeq2i 5040 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
 |-  A  =  B   =>    |-  ( F : A
 --> C  <->  F : B --> C )
 
Theoremfeq23i 5041 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  C   &    |-  B  =  D   =>    |-  ( F : A --> B 
 <->  F : C --> D )
 
Theoremfeq23d 5042 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
 |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( F : A --> B  <->  F : C --> D ) )
 
Theoremnff 5043 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x F   &    |-  F/_ x A   &    |-  F/_ x B   =>    |- 
 F/ x  F : A
 --> B
 
Theoremsbcfng 5044* Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
 
Theoremsbcfg 5045* Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  ( X  e.  V  ->  ( [. X  /  x ]. F : A --> B 
 <-> 
 [_ X  /  x ]_ F : [_ X  /  x ]_ A --> [_ X  /  x ]_ B ) )
 
Theoremffn 5046 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  F  Fn  A )
 
Theoremdffn2 5047 Any function is a mapping into  _V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F  Fn  A  <->  F : A --> _V )
 
Theoremffun 5048 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Fun  F )
 
Theoremfrel 5049 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Rel  F )
 
Theoremfdm 5050 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  dom  F  =  A )
 
Theoremfdmi 5051 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
 |-  F : A --> B   =>    |-  dom  F  =  A
 
Theoremfrn 5052 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  ran  F  C_  B )
 
Theoremdffn3 5053 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
 |-  ( F  Fn  A  <->  F : A --> ran  F )
 
Theoremfss 5054 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F : A
 --> B  /\  B  C_  C )  ->  F : A
 --> C )
 
Theoremfssd 5055 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremfco 5056 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F : B
 --> C  /\  G : A
 --> B )  ->  ( F  o.  G ) : A --> C )
 
Theoremfco2 5057 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
 
Theoremfssxp 5058 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
 
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.)
 |-  ( ( F : A
 --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
 
Theoremfunssxp 5060 Two ways of specifying a partial function from  A to  B. (Contributed by NM, 13-Nov-2007.)
 |-  ( ( Fun  F  /\  F  C_  ( A  X.  B ) )  <->  ( F : dom  F --> B  /\  dom  F 
 C_  A ) )
 
Theoremffdm 5061 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
 |-  ( F : A --> B  ->  ( F : dom  F --> B  /\  dom  F 
 C_  A ) )
 
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.)
 |-  ( ( F : A
 --> B  /\  <. C ,  D >.  e.  F ) 
 ->  ( C  e.  A  /\  D  e.  B ) )
 
Theoremfun 5063 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
 |-  ( ( ( F : A --> C  /\  G : B --> D ) 
 /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B ) --> ( C  u.  D ) )
 
Theoremfun2 5064 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( ( F : A --> C  /\  G : B --> C ) 
 /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B ) --> C )
 
Theoremfnfco 5065 Composition of two functions. (Contributed by NM, 22-May-2006.)
 |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B )
 
Theoremfssres 5066 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
 |-  ( ( F : A
 --> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
 
Theoremfssres2 5067 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
 |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
 
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.)
 |-  ( F : A --> B  ->  ( F  |`  X ) : ( A  i^i  X ) --> B )
 
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.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  ( ( F  |`  ( A  i^i  B ) )  u.  (
 ( F  |`  ( A 
 \  B ) )  u.  ( G  |`  ( B 
 \  A ) ) ) )  C_  ( F  u.  G ) )
 
Theoremfcoi1 5070 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F : A --> B  ->  ( F  o.  (  _I  |`  A )
 )  =  F )
 
Theoremfcoi2 5071 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F : A --> B  ->  ( (  _I  |`  B )  o.  F )  =  F )
 
Theoremfeu 5072* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
 |-  ( ( F : A
 --> B  /\  C  e.  A )  ->  E! y  e.  B  <. C ,  y >.  e.  F )
 
Theoremfcnvres 5073 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
 |-  ( F : A --> B  ->  `' ( F  |`  A )  =  ( `' F  |`  B ) )
 
Theoremfimacnvdisj 5074 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
 |-  ( ( F : A
 --> B  /\  ( B  i^i  C )  =  (/) )  ->  ( `' F " C )  =  (/) )
 
Theoremfintm 5075* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
 |- 
 E. x  x  e.  B   =>    |-  ( F : A --> |^|
 B 
 <-> 
 A. x  e.  B  F : A --> x )
 
Theoremfin 5076 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F : A --> ( B  i^i  C )  <-> 
 ( F : A --> B  /\  F : A --> C ) )
 
Theoremfabexg 5077* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  F  =  { x  |  ( x : A --> B  /\  ph ) }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
 
Theoremfabex 5078* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  F  =  { x  |  ( x : A --> B  /\  ph ) }   =>    |-  F  e.  _V
 
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.)
 |-  ( ( F  e.  C  /\  F : A --> B )  ->  A  e.  _V )
 
Theoremf0 5080 The empty function. (Contributed by NM, 14-Aug-1999.)
 |-  (/) : (/) --> A
 
Theoremf00 5081 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
 |-  ( F : A --> (/)  <->  ( F  =  (/)  /\  A  =  (/) ) )
 
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.)
 |-  B  e.  _V   =>    |-  ( A  X.  { B } ) : A --> { B }
 
Theoremfconstg 5083 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
 |-  ( B  e.  V  ->  ( A  X.  { B } ) : A --> { B } )
 
Theoremfnconstg 5084 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
 |-  ( B  e.  V  ->  ( A  X.  { B } )  Fn  A )
 
Theoremfconst6g 5085 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( B  e.  C  ->  ( A  X.  { B } ) : A --> C )
 
Theoremfconst6 5086 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
 |-  B  e.  C   =>    |-  ( A  X.  { B } ) : A --> C
 
Theoremf1eq1 5087 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( F  =  G  ->  ( F : A -1-1-> B  <->  G : A -1-1-> B ) )
 
Theoremf1eq2 5088 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( A  =  B  ->  ( F : A -1-1-> C  <->  F : B -1-1-> C ) )
 
Theoremf1eq3 5089 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )
 
Theoremnff1 5090 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
 |-  F/_ x F   &    |-  F/_ x A   &    |-  F/_ x B   =>    |- 
 F/ x  F : A -1-1-> B
 
Theoremdff12 5091* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
 |-  ( F : A -1-1-> B  <-> 
 ( F : A --> B  /\  A. y E* x  x F y ) )
 
Theoremf1f 5092 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
 |-  ( F : A -1-1-> B 
 ->  F : A --> B )
 
Theoremf1fn 5093 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  F  Fn  A )
 
Theoremf1fun 5094 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  Fun  F )
 
Theoremf1rel 5095 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  Rel  F )
 
Theoremf1dm 5096 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  dom  F  =  A )
 
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.)
 |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A -1-1-> C )
 
Theoremf1ssr 5098 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C )  ->  F : A -1-1-> C )
 
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.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
 
Theoremf1cnvcnv 5100 Two ways to express that a set  A (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.)
 |-  ( `' `' A : dom  A -1-1-> _V  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
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