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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfcnvres 5101 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
 |-  ( F : A --> B  ->  `' ( F  |`  A )  =  ( `' F  |`  B ) )
 
Theoremfimacnvdisj 5102 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 5103* 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 5104 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 5105* 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 5106* 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 5107 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 5108 The empty function. (Contributed by NM, 14-Aug-1999.)
 |-  (/) : (/) --> A
 
Theoremf00 5109 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 5110 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 5111 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 5112 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 5113 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( B  e.  C  ->  ( A  X.  { B } ) : A --> C )
 
Theoremfconst6 5114 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 5115 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 5116 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 5117 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 5118 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 5119* 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 5120 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
 |-  ( F : A -1-1-> B 
 ->  F : A --> B )
 
Theoremf1fn 5121 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 5122 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  Fun  F )
 
Theoremf1rel 5123 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  Rel  F )
 
Theoremf1dm 5124 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  dom  F  =  A )
 
Theoremf1ss 5125 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 5126 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 5127 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 5128 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 ) )
 
Theoremf1co 5129 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)
 |-  ( ( F : B -1-1-> C  /\  G : A -1-1-> B )  ->  ( F  o.  G ) : A -1-1-> C )
 
Theoremfoeq1 5130 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F : A -onto-> B 
 <->  G : A -onto-> B ) )
 
Theoremfoeq2 5131 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : A -onto-> C 
 <->  F : B -onto-> C ) )
 
Theoremfoeq3 5132 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : C -onto-> A 
 <->  F : C -onto-> B ) )
 
Theoremnffo 5133 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
 |-  F/_ x F   &    |-  F/_ x A   &    |-  F/_ x B   =>    |- 
 F/ x  F : A -onto-> B
 
Theoremfof 5134 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A -onto-> B  ->  F : A --> B )
 
Theoremfofun 5135 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
 |-  ( F : A -onto-> B  ->  Fun  F )
 
Theoremfofn 5136 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
 |-  ( F : A -onto-> B  ->  F  Fn  A )
 
Theoremforn 5137 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A -onto-> B  ->  ran  F  =  B )
 
Theoremdffo2 5138 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  ran  F  =  B ) )
 
Theoremfoima 5139 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
 |-  ( F : A -onto-> B  ->  ( F " A )  =  B )
 
Theoremdffn4 5140 A function maps onto its range. (Contributed by NM, 10-May-1998.)
 |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
 
Theoremfunforn 5141 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
 |-  ( Fun  A  <->  A : dom  A -onto-> ran  A )
 
Theoremfodmrnu 5142 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
 |-  ( ( F : A -onto-> B  /\  F : C -onto-> D )  ->  ( A  =  C  /\  B  =  D )
 )
 
Theoremfores 5143 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F  |`  A ) : A -onto-> ( F
 " A ) )
 
Theoremfoco 5144 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
 |-  ( ( F : B -onto-> C  /\  G : A -onto-> B )  ->  ( F  o.  G ) : A -onto-> C )
 
Theoremf1oeq1 5145 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( F  =  G  ->  ( F : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
 
Theoremf1oeq2 5146 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
 
Theoremf1oeq3 5147 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( A  =  B  ->  ( F : C -1-1-onto-> A  <->  F : C -1-1-onto-> B ) )
 
Theoremf1oeq23 5148 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( F : A
 -1-1-onto-> C 
 <->  F : B -1-1-onto-> D ) )
 
Theoremf1eq123d 5149 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( F : A -1-1-> C  <->  G : B -1-1-> D ) )
 
Theoremfoeq123d 5150 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( F : A -onto-> C  <->  G : B -onto-> D ) )
 
Theoremf1oeq123d 5151 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
 
Theoremnff1o 5152 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
 |-  F/_ x F   &    |-  F/_ x A   &    |-  F/_ x B   =>    |- 
 F/ x  F : A
 -1-1-onto-> B
 
Theoremf1of1 5153 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
 
Theoremf1of 5154 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  F : A --> B )
 
Theoremf1ofn 5155 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  F  Fn  A )
 
Theoremf1ofun 5156 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  Fun  F )
 
Theoremf1orel 5157 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  Rel  F )
 
Theoremf1odm 5158 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
 
Theoremdff1o2 5159 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
 
Theoremdff1o3 5160 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
 
Theoremf1ofo 5161 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
 |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
 
Theoremdff1o4 5162 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
 
Theoremdff1o5 5163 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
 
Theoremf1orn 5164 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
 |-  ( F : A -1-1-onto-> ran  F  <-> 
 ( F  Fn  A  /\  Fun  `' F ) )
 
Theoremf1f1orn 5165 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
 |-  ( F : A -1-1-> B 
 ->  F : A -1-1-onto-> ran  F )
 
Theoremf1oabexg 5166* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  F  =  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
 
Theoremf1ocnv 5167 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 : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
 
Theoremf1ocnvb 5168 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 : A -1-1-onto-> B  <->  `' F : B -1-1-onto-> A ) )
 
Theoremf1ores 5169 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
 
Theoremf1orescnv 5170 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 5171 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' F " ( F
 " C ) )  =  C )
 
Theoremfoimacnv 5172 A reverse version of f1imacnv 5171. (Contributed by Jeff Hankins, 16-Jul-2009.)
 |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( F
 " ( `' F " C ) )  =  C )
 
Theoremfoun 5173 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
 |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D ) 
 /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  C ) -onto-> ( B  u.  D ) )
 
Theoremf1oun 5174 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
 |-  ( ( ( F : A -1-1-onto-> B  /\  G : C
 -1-1-onto-> D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( F  u.  G ) : ( A  u.  C )
 -1-1-onto-> ( B  u.  D ) )
 
Theoremfun11iun 5175* 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.)
 |-  ( x  =  y 
 ->  B  =  C )   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  ( B : D -1-1-> S  /\  A. y  e.  A  ( B  C_  C  \/  C  C_  B ) )  ->  U_ x  e.  A  B : U_ x  e.  A  D -1-1-> S )
 
Theoremresdif 5176 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  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D ) 
 ->  ( F  |`  ( A 
 \  B ) ) : ( A  \  B ) -1-1-onto-> ( C  \  D ) )
 
Theoremf1oco 5177 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
 |-  ( ( F : B
 -1-1-onto-> C  /\  G : A -1-1-onto-> B )  ->  ( F  o.  G ) : A -1-1-onto-> C )
 
Theoremf1cnv 5178 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
 |-  ( F : A -1-1-> B 
 ->  `' F : ran  F -1-1-onto-> A )
 
Theoremfuncocnv2 5179 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( Fun  F  ->  ( F  o.  `' F )  =  (  _I  |` 
 ran  F ) )
 
Theoremfococnv2 5180 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  ( F : A -onto-> B  ->  ( F  o.  `' F )  =  (  _I  |`  B )
 )
 
Theoremf1ococnv2 5181 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 : A -1-1-onto-> B  ->  ( F  o.  `' F )  =  (  _I  |`  B ) )
 
Theoremf1cocnv2 5182 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
 |-  ( F : A -1-1-> B 
 ->  ( F  o.  `' F )  =  (  _I  |`  ran  F )
 )
 
Theoremf1ococnv1 5183 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 : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
 
Theoremf1cocnv1 5184 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
 |-  ( F : A -1-1-> B 
 ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
 
Theoremfuncoeqres 5185 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 ) )
 
Theoremffoss 5186* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
 |-  F  e.  _V   =>    |-  ( F : A
 --> B  <->  E. x ( F : A -onto-> x  /\  x  C_  B ) )
 
Theoremf11o 5187* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
 |-  F  e.  _V   =>    |-  ( F : A -1-1-> B  <->  E. x ( F : A -1-1-onto-> x  /\  x  C_  B ) )
 
Theoremf10 5188 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
 |-  (/) : (/) -1-1-> A
 
Theoremf1o00 5189 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
 |-  ( F : (/) -1-1-onto-> A  <->  ( F  =  (/)  /\  A  =  (/) ) )
 
Theoremfo00 5190 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
 |-  ( F : (/) -onto-> A  <-> 
 ( F  =  (/)  /\  A  =  (/) ) )
 
Theoremf1o0 5191 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
 |-  (/) : (/)
 -1-1-onto-> (/)
 
Theoremf1oi 5192 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  (  _I  |`  A ) : A -1-1-onto-> A
 
Theoremf1ovi 5193 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
 |- 
 _I  : _V -1-1-onto-> _V
 
Theoremf1osn 5194 A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { <. A ,  B >. } : { A }
 -1-1-onto-> { B }
 
Theoremf1osng 5195 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. } : { A } -1-1-onto-> { B } )
 
Theoremf1oprg 5196 An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
 )  ->  ( ( A  =/=  C  /\  B  =/=  D )  ->  { <. A ,  B >. ,  <. C ,  D >. } : { A ,  C } -1-1-onto-> { B ,  D }
 ) )
 
Theoremtz6.12-2 5197* Function value when  F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
 
Theoremfveu 5198* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
 |-  ( E! x  A F x  ->  ( F `
  A )  = 
 U. { x  |  A F x } )
 
Theorembrprcneu 5199* If  A is a proper class, then there is no unique binary relationship with  A as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)
 |-  ( -.  A  e.  _V 
 ->  -.  E! x  A F x )
 
Theoremfvprc 5200 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
 |-  ( -.  A  e.  _V 
 ->  ( F `  A )  =  (/) )
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