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Theorem List for Metamath Proof Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfsnunres 5901 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  ( ( F  Fn  S  /\  -.  X  e.  S )  ->  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  F )
 
Theoremfvpr1 5902 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
 |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
 
Theoremfvpr2 5903 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
 |-  B  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
 
Theoremfvpr1g 5904 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  ( ( A  e.  V  /\  C  e.  W  /\  A  =/=  B ) 
 ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
 
Theoremfvpr2g 5905 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  ( ( B  e.  V  /\  D  e.  W  /\  A  =/=  B ) 
 ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
 
Theoremfvtp1 5906 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  A  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( A  =/=  B 
 /\  A  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  A )  =  D )
 
Theoremfvtp2 5907 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  B  e.  _V   &    |-  E  e.  _V   =>    |-  ( ( A  =/=  B 
 /\  B  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B )  =  E )
 
Theoremfvtp3 5908 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  C  e.  _V   &    |-  F  e.  _V   =>    |-  ( ( A  =/=  C 
 /\  B  =/=  C )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C )  =  F )
 
Theoremfvtp1g 5909 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( A  e.  V  /\  D  e.  W )  /\  ( A  =/=  B  /\  A  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  A )  =  D )
 
Theoremfvtp2g 5910 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( B  e.  V  /\  E  e.  W )  /\  ( A  =/=  B  /\  B  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  B )  =  E )
 
Theoremfvtp3g 5911 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( C  e.  V  /\  F  e.  W )  /\  ( A  =/=  C  /\  B  =/=  C ) )  ->  ( { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } `  C )  =  F )
 
Theoremfvconst2g 5912 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
 |-  ( ( B  e.  D  /\  C  e.  A )  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2g 5913 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
 |-  ( B  e.  C  ->  ( F : A --> { B }  <->  F  =  ( A  X.  { B }
 ) ) )
 
Theoremfvconst2 5914 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
 |-  B  e.  _V   =>    |-  ( C  e.  A  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2 5915 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
 |-  B  e.  _V   =>    |-  ( F : A
 --> { B }  <->  F  =  ( A  X.  { B }
 ) )
 
Theoremfconst5 5916 Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)
 |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( F  =  ( A  X.  { B }
 ) 
 <-> 
 ran  F  =  { B } ) )
 
Theoremfnpr 5917 Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) (Revised by NM, 10-Dec-2017.)
 |-  I  e.  _V   &    |-  J  e.  _V   =>    |-  ( I  =/=  J  ->  ( F  Fn  { I ,  J }  <->  F  =  { <. I ,  ( F `  I )
 >. ,  <. J ,  ( F `  J ) >. } ) )
 
TheoremfnprOLD 5918 Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Obsolete version of fnpr 5917 as of 10-Dec-2017. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  I  e.  A   &    |-  J  e.  B   =>    |-  ( I  =/=  J  ->  ( F  Fn  { I ,  J }  <->  F  =  { <. I ,  ( F `  I )
 >. ,  <. J ,  ( F `  J ) >. } ) )
 
Theoremfnsuppres 5919 Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  (
 ( `' F "
 ( _V  \  { Z } ) )  C_  A 
 <->  ( F  |`  B )  =  ( B  X.  { Z } ) ) )
 
Theoremfnsuppeq0 5920 The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.)
 |-  ( ( F  Fn  A  /\  Z  e.  V )  ->  ( ( `' F " ( _V  \  { Z } )
 )  =  (/)  <->  F  =  ( A  X.  { Z }
 ) ) )
 
Theoremfconstfv 5921* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5915. (Contributed by NM, 27-Aug-2004.)
 |-  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
 
Theoremfconst3 5922 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
 |-  ( F : A --> { B }  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
 
Theoremfconst4 5923 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
 |-  ( F : A --> { B }  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
 
Theoremresfunexg 5924 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
TheoremresfunexgALT 5925 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5924 but requires ax-pow 4345. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
Theoremcofunexg 5926 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  o.  B )  e.  _V )
 
Theoremcofunex2g 5927 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
 |-  ( ( A  e.  V  /\  Fun  `' B )  ->  ( A  o.  B )  e.  _V )
 
Theoremfnex 5928 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5924. See fnexALT 5929 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
 
TheoremfnexALT 5929 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5497. This version of fnex 5928 uses ax-pow 4345, whereas fnex 5928 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
 
Theoremfunex 5930 If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5928. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
 |-  ( ( Fun  F  /\  dom  F  e.  B )  ->  F  e.  _V )
 
Theoremopabex 5931* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  E* y ph )   =>    |-  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremmptexg 5932* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  V  ->  ( x  e.  A  |->  B )  e.  _V )
 
Theoremmptex 5933* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  A  e.  _V   =>    |-  ( x  e.  A  |->  B )  e. 
 _V
 
Theoremfunrnex 5934 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5930. (Contributed by NM, 11-Nov-1995.)
 |-  ( dom  F  e.  B  ->  ( Fun  F  ->  ran  F  e.  _V ) )
 
Theoremzfrep6 5935* A version of the Axiom of Replacement. Normally  ph would have free variables  x and  y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4298 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 4288. (Contributed by NM, 10-Oct-2003.)
 |-  ( A. x  e.  z  E! y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
 
Theoremfex 5936 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
 |-  ( ( F : A
 --> B  /\  A  e.  C )  ->  F  e.  _V )
 
Theoremfornex 5937 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  e.  _V ) )
 
Theoremf1dmex 5938 If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 4288. (Contributed by NM, 4-Sep-2004.)
 |-  ( ( F : A -1-1-> B  /\  B  e.  C )  ->  A  e.  _V )
 
Theoremeufnfv 5939* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 E! f ( f  Fn  A  /\  A. x  e.  A  (
 f `  x )  =  B )
 
Theoremfunfvima 5940 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
 |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) )
 
Theoremfunfvima2 5941 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) )
 
Theoremfunfvima3 5942 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
 |-  ( ( Fun  F  /\  F  C_  G )  ->  ( A  e.  dom  F 
 ->  ( F `  A )  e.  ( G " { A } )
 ) )
 
Theoremfnfvima 5943 The function value of an operand in a set is contained in the image of that set, using the  Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
 |-  ( ( F  Fn  A  /\  S  C_  A  /\  X  e.  S ) 
 ->  ( F `  X )  e.  ( F " S ) )
 
Theoremrexima 5944* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( E. x  e.  ( F " B ) ph  <->  E. y  e.  B  ps ) )
 
Theoremralima 5945* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( A. x  e.  ( F " B ) ph  <->  A. y  e.  B  ps ) )
 
Theoremidref 5946* TODO: This is the same as issref 5214 (which has a much longer proof). Should we replace issref 5214 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

 |-  ( (  _I  |`  A ) 
 C_  R  <->  A. x  e.  A  x R x )
 
Theoremfvclss 5947* Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)
 |- 
 { y  |  E. x  y  =  ( F `  x ) }  C_  ( ran  F  u.  { (/) } )
 
Theoremfvclex 5948* Existence of the class of values of a set. (Contributed by NM, 9-Nov-1995.)
 |-  F  e.  _V   =>    |-  { y  | 
 E. x  y  =  ( F `  x ) }  e.  _V
 
Theoremfvresex 5949* Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  A  e.  _V   =>    |-  { y  | 
 E. x  y  =  ( ( F  |`  A ) `
  x ) }  e.  _V
 
Theoremabrexex 5950* Existence of a class abstraction of existentially restricted sets.  x is normally a free-variable parameter in the class expression substituted for  B, which can be thought of as  B ( x ). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5932, funex 5930, fnex 5928, resfunexg 5924, and funimaexg 5497. See also abrexex2 5968. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   =>    |-  { y  | 
 E. x  e.  A  y  =  B }  e.  _V
 
Theoremabrexexg 5951* Existence of a class abstraction of existentially restricted sets.  x is normally a free-variable parameter in  B. The antecedent assures us that  A is a set. (Contributed by NM, 3-Nov-2003.)
 |-  ( A  e.  V  ->  { y  |  E. x  e.  A  y  =  B }  e.  _V )
 
Theoremelabrex 5952* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
 |-  B  e.  _V   =>    |-  ( x  e.  A  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
 
Theoremabrexco 5953* Composition of two image maps  C ( y ) and 
B ( w ). (Contributed by NM, 27-May-2013.)
 |-  B  e.  _V   &    |-  (
 y  =  B  ->  C  =  D )   =>    |-  { x  |  E. y  e.  { z  |  E. w  e.  A  z  =  B } x  =  C }  =  { x  |  E. w  e.  A  x  =  D }
 
Theoremiunexg 5954* The existence of an indexed union. 
x is normally a free-variable parameter in  B. (Contributed by NM, 23-Mar-2006.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  W )  ->  U_ x  e.  A  B  e.  _V )
 
Theoremabrexex2g 5955* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  { y  | 
 ph }  e.  W )  ->  { y  | 
 E. x  e.  A  ph
 }  e.  _V )
 
Theoremopabex3d 5956* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  { y  |  ps }  e.  _V )   =>    |-  ( ph  ->  {
 <. x ,  y >.  |  ( x  e.  A  /\  ps ) }  e.  _V )
 
Theoremopabex3 5957* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  A  e.  _V   &    |-  ( x  e.  A  ->  { y  |  ph }  e.  _V )   =>    |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremiunex 5958* The existence of an indexed union. 
x is normally a free-variable parameter in the class expression substituted for  B, which can be read informally as  B ( x ). (Contributed by NM, 13-Oct-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  U_ x  e.  A  B  e.  _V
 
Theoremimaiun 5959* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
 |-  ( A " U_ x  e.  B  C )  = 
 U_ x  e.  B  ( A " C )
 
Theoremimauni 5960* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
 |-  ( A " U. B )  =  U_ x  e.  B  ( A " x )
 
Theoremfniunfv 5961* The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
 |-  ( F  Fn  A  -> 
 U_ x  e.  A  ( F `  x )  =  U. ran  F )
 
Theoremfuniunfv 5962* The indexed union of a function's values is the union of its image under the index class.

Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to  F  Fn  A, the theorem can be proved without this dependency. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

 |-  ( Fun  F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F
 " A ) )
 
Theoremfuniunfvf 5963* The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5962 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.)
 |-  F/_ x F   =>    |-  ( Fun  F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F
 " A ) )
 
Theoremeluniima 5964* Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)
 |-  ( Fun  F  ->  ( B  e.  U. ( F " A )  <->  E. x  e.  A  B  e.  ( F `  x ) ) )
 
Theoremelunirn 5965* Membership in the union of the range of a function. See elunirnALT 5967 for alternate proof. (Contributed by NM, 24-Sep-2006.)
 |-  ( Fun  F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `
  x ) ) )
 
Theoremfnunirn 5966* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( F  Fn  I  ->  ( A  e.  U. ran  F  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
 
TheoremelunirnALT 5967* Membership in the union of the range of a function, proved directly. Unlike elunirn 5965, it doesn't appeal to ndmfv 5722 (via funiunfv 5962). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Fun  F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `
  x ) ) )
 
Theoremabrexex2 5968* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 5950. (Contributed by NM, 12-Sep-2004.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x  e.  A  ph
 }  e.  _V
 
Theoremabexssex 5969* Existence of a class abstraction with an existentially quantified expression. Both  x and  y can be free in  ph. (Contributed by NM, 29-Jul-2006.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x ( x 
 C_  A  /\  ph ) }  e.  _V
 
Theoremabexex 5970* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
 |-  A  e.  _V   &    |-  ( ph  ->  x  e.  A )   &    |- 
 { y  |  ph }  e.  _V   =>    |- 
 { y  |  E. x ph }  e.  _V
 
Theoremdff13 5971* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)
 |-  ( F : A -1-1-> B  <-> 
 ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  ( ( F `
  x )  =  ( F `  y
 )  ->  x  =  y ) ) )
 
Theoremf1veqaeq 5972 If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( ( F `  C )  =  ( F `  D )  ->  C  =  D )
 )
 
Theoremdff13f 5973* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)
 |-  F/_ x F   &    |-  F/_ y F   =>    |-  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  A  ( ( F `
  x )  =  ( F `  y
 )  ->  x  =  y ) ) )
 
Theoremf1mpt 5974* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( x  =  y  ->  C  =  D )   =>    |-  ( F : A -1-1-> B  <->  (
 A. x  e.  A  C  e.  B  /\  A. x  e.  A  A. y  e.  A  ( C  =  D  ->  x  =  y ) ) )
 
Theoremf1fveq 5975 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)
 |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( ( F `  C )  =  ( F `  D )  <->  C  =  D ) )
 
Theoremf1elima 5976 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( F : A -1-1-> B  /\  X  e.  A  /\  Y  C_  A )  ->  ( ( F `
  X )  e.  ( F " Y ) 
 <->  X  e.  Y ) )
 
Theoremf1imass 5977 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ( F : A -1-1-> B  /\  ( C 
 C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C_  ( F
 " D )  <->  C  C_  D ) )
 
Theoremf1imaeq 5978 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ( F : A -1-1-> B  /\  ( C 
 C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  =  ( F " D )  <->  C  =  D ) )
 
Theoremf1imapss 5979 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
 |-  ( ( F : A -1-1-> B  /\  ( C 
 C_  A  /\  D  C_  A ) )  ->  ( ( F " C )  C.  ( F
 " D )  <->  C  C.  D ) )
 
Theoremdff1o6 5980* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
 |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  ran 
 F  =  B  /\  A. x  e.  A  A. y  e.  A  (
 ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
 
Theoremf1ocnvfv1 5981 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  A )  ->  ( `' F `  ( F `  C ) )  =  C )
 
Theoremf1ocnvfv2 5982 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  ( F `  ( `' F `  C ) )  =  C )
 
Theoremf1ocnvfv 5983 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  A )  ->  ( ( F `
  C )  =  D  ->  ( `' F `  D )  =  C ) )
 
Theoremf1ocnvfvb 5984 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B ) 
 ->  ( ( F `  C )  =  D  <->  ( `' F `  D )  =  C ) )
 
Theoremf1ocnvdm 5985 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A )
 
Theoremf1ocnvfvrneq 5986 If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  ( ( F : A -1-1-> B  /\  ( C  e.  ran  F  /\  D  e.  ran  F ) )  ->  ( ( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) )
 
Theoremfcof1 5987 An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  ( ( F : A
 --> B  /\  ( R  o.  F )  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
 
Theoremfcofo 5988 An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  ( ( F : A
 --> B  /\  S : B
 --> A  /\  ( F  o.  S )  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
 
Theoremcbvfo 5989* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( F `  x )  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( F : A -onto-> B  ->  ( A. x  e.  A  ph  <->  A. y  e.  B  ps ) )
 
Theoremcbvexfo 5990* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
 |-  ( ( F `  x )  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( F : A -onto-> B  ->  ( E. x  e.  A  ph  <->  E. y  e.  B  ps ) )
 
Theoremcocan1 5991 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( F : B -1-1-> C  /\  H : A
 --> B  /\  K : A
 --> B )  ->  (
 ( F  o.  H )  =  ( F  o.  K )  <->  H  =  K ) )
 
Theoremcocan2 5992 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B )  ->  ( ( H  o.  F )  =  ( K  o.  F ) 
 <->  H  =  K ) )
 
Theoremfcof1o 5993 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( ( F : A --> B  /\  G : B --> A ) 
 /\  ( ( F  o.  G )  =  (  _I  |`  B ) 
 /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F : A -1-1-onto-> B  /\  `' F  =  G ) )
 
Theoremfoeqcnvco 5994 Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  ( ( F : A -onto-> B  /\  G : A -onto-> B )  ->  ( F  =  G  <->  ( F  o.  `' G )  =  (  _I  |`  B )
 ) )
 
Theoremf1eqcocnv 5995 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  ( ( F : A -1-1-> B  /\  G : A -1-1-> B )  ->  ( F  =  G  <->  ( `' F  o.  G )  =  (  _I  |`  A )
 ) )
 
Theoremfveqf1o 5996 Given a bijection  F, produce another bijection  G which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  G  =  ( F  o.  ( (  _I  |`  ( A  \  { C ,  ( `' F `  D ) }
 ) )  u.  { <. C ,  ( `' F `  D )
 >. ,  <. ( `' F `  D ) ,  C >. } ) )   =>    |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( G : A
 -1-1-onto-> B  /\  ( G `  C )  =  D ) )
 
Theoremfliftrel 5997*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  F  C_  ( R  X.  S ) )
 
Theoremfliftel 5998* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B ) ) )
 
Theoremfliftel1 5999* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ( ph  /\  x  e.  X )  ->  A F B )
 
Theoremfliftcnv 6000* Converse of the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  `' F  =  ran  ( x  e.  X  |->  <. B ,  A >. ) )
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