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Theorem List for Intuitionistic Logic Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremressnop0 5601 If  A is not in  C, then the restriction of a singleton of  <. A ,  B >. to  C is null. (Contributed by Scott Fenton, 15-Apr-2011.)
 |-  ( -.  A  e.  C  ->  ( { <. A ,  B >. }  |`  C )  =  (/) )
 
Theoremfpr 5602 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  { <. A ,  C >. ,  <. B ,  D >. } : { A ,  B } --> { C ,  D } )
 
Theoremfprg 5603 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
 |-  ( ( ( A  e.  E  /\  B  e.  F )  /\  ( C  e.  G  /\  D  e.  H )  /\  A  =/=  B ) 
 ->  { <. A ,  C >. ,  <. B ,  D >. } : { A ,  B } --> { C ,  D } )
 
Theoremftpg 5604 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  /\  ( A  e.  F  /\  B  e.  G  /\  C  e.  H )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  { <. X ,  A >. ,  <. Y ,  B >. ,  <. Z ,  C >. } : { X ,  Y ,  Z } --> { A ,  B ,  C }
 )
 
Theoremftp 5605 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   &    |-  A  =/=  B   &    |-  A  =/=  C   &    |-  B  =/=  C   =>    |-  { <. A ,  X >. ,  <. B ,  Y >. ,  <. C ,  Z >. } : { A ,  B ,  C } --> { X ,  Y ,  Z }
 
Theoremfnressn 5606 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
  B ) >. } )
 
Theoremfressnfv 5607 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C ) )
 
Theoremfvconst 5608 The value of a constant function. (Contributed by NM, 30-May-1999.)
 |-  ( ( F : A
 --> { B }  /\  C  e.  A )  ->  ( F `  C )  =  B )
 
Theoremfmptsn 5609* Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
 
Theoremfmptap 5610* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( R  u.  { A } )  =  S   &    |-  ( x  =  A  ->  C  =  B )   =>    |-  ( ( x  e.  R  |->  C )  u. 
 { <. A ,  B >. } )  =  ( x  e.  S  |->  C )
 
Theoremfmptapd 5611* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( R  u.  { A } )  =  S )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  C  =  B )   =>    |-  ( ph  ->  ( ( x  e.  R  |->  C )  u.  { <. A ,  B >. } )  =  ( x  e.  S  |->  C ) )
 
Theoremfmptpr 5612* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  E  =  C )   &    |-  ( ( ph  /\  x  =  B )  ->  E  =  D )   =>    |-  ( ph  ->  { <. A ,  C >. ,  <. B ,  D >. }  =  ( x  e.  { A ,  B }  |->  E ) )
 
Theoremfvresi 5613 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)
 |-  ( B  e.  A  ->  ( (  _I  |`  A ) `
  B )  =  B )
 
Theoremfvunsng 5614 Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.)
 |-  ( ( D  e.  V  /\  B  =/=  D )  ->  ( ( A  u.  { <. B ,  C >. } ) `  D )  =  ( A `  D ) )
 
Theoremfvsn 5615 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { <. A ,  B >. } `  A )  =  B
 
Theoremfvsng 5616 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { <. A ,  B >. } `  A )  =  B )
 
Theoremfvsnun1 5617 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5618. (Contributed by NM, 23-Sep-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  G  =  ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )   =>    |-  ( G `  A )  =  B
 
Theoremfvsnun2 5618 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5617. (Contributed by NM, 23-Sep-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  G  =  ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )   =>    |-  ( D  e.  ( C  \  { A }
 )  ->  ( G `  D )  =  ( F `  D ) )
 
Theoremfnsnsplitss 5619 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)
 |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( F  |`  ( A  \  { X } ) )  u. 
 { <. X ,  ( F `  X ) >. } )  C_  F )
 
Theoremfsnunf 5620 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  ( ( F : S
 --> T  /\  ( X  e.  V  /\  -.  X  e.  S )  /\  Y  e.  T ) 
 ->  ( F  u.  { <. X ,  Y >. } ) : ( S  u.  { X }
 ) --> T )
 
Theoremfsnunfv 5621 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
 |-  ( ( X  e.  V  /\  Y  e.  W  /\  -.  X  e.  dom  F )  ->  ( ( F  u.  { <. X ,  Y >. } ) `  X )  =  Y )
 
Theoremfsnunres 5622 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 )
 
Theoremfunresdfunsnss 5623 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)
 |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  ( ( F  |`  ( _V  \  { X } ) )  u. 
 { <. X ,  ( F `  X ) >. } )  C_  F )
 
Theoremfvpr1 5624 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 5625 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 5626 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 5627 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 )
 
Theoremfvtp1g 5628 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 5629 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 5630 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 )
 
Theoremfvtp1 5631 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 5632 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 5633 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 )
 
Theoremfvconst2g 5634 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
 |-  ( ( B  e.  D  /\  C  e.  A )  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2g 5635 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 5636 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
 |-  B  e.  _V   =>    |-  ( C  e.  A  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2 5637 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
 |-  B  e.  _V   =>    |-  ( F : A
 --> { B }  <->  F  =  ( A  X.  { B }
 ) )
 
Theoremfconstfvm 5638* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5637. (Contributed by Jim Kingdon, 8-Jan-2019.)
 |-  ( E. y  y  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) ) )
 
Theoremfconst3m 5639* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
 |-  ( E. x  x  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  A  C_  ( `' F " { B }
 ) ) ) )
 
Theoremfconst4m 5640* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
 |-  ( E. x  x  e.  A  ->  ( F : A --> { B } 
 <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) ) )
 
Theoremresfunexg 5641 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 )
 
Theoremfnex 5642 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 5641. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
 
Theoremfunex 5643 If the domain of a function exists, so does 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 5642. (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 5644* 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 5645* 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 5646* 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
 
Theoremfex 5647 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 )
 
Theoremeufnfv 5648* 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 5649 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 5650 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 5651 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 5652 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 ) )
 
Theoremfoima2 5653* Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5350). (Contributed by BJ, 6-Jul-2022.)
 |-  ( F : A -onto-> B  ->  ( Y  e.  B 
 <-> 
 E. x  e.  A  Y  =  ( F `  x ) ) )
 
Theoremfoelrn 5654* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.)
 |-  ( ( F : A -onto-> B  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
 
Theoremfoco2 5655 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  ( ( F : B
 --> C  /\  G : A
 --> B  /\  ( F  o.  G ) : A -onto-> C )  ->  F : B -onto-> C )
 
Theoremrexima 5656* 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 5657* 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 5658* TODO: This is the same as issref 4921 (which has a much longer proof). Should we replace issref 4921 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 )
 
Theoremelabrex 5659* 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 5660* 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 }
 
Theoremimaiun 5661* 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 5662* 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 5663* 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 )
 
Theoremfuniunfvdm 5664* The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5663. (Contributed by Jim Kingdon, 10-Jan-2019.)
 |-  ( F  Fn  A  -> 
 U_ x  e.  A  ( F `  x )  =  U. ( F
 " A ) )
 
Theoremfuniunfvdmf 5665* The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5664 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
 |-  F/_ x F   =>    |-  ( F  Fn  A  -> 
 U_ x  e.  A  ( F `  x )  =  U. ( F
 " A ) )
 
Theoremeluniimadm 5666* Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
 |-  ( F  Fn  A  ->  ( B  e.  U. ( F " A )  <->  E. x  e.  A  B  e.  ( F `  x ) ) )
 
Theoremelunirn 5667* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
 |-  ( Fun  F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `
  x ) ) )
 
Theoremfnunirn 5668* 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 ) ) )
 
Theoremdff13 5669* 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 5670 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 5671* 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 5672* 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 5673 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 5674 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 5675 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 5676 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 ) )
 
Theoremdff1o6 5677* 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 5678 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 5679 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 5680 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 5681 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 5682 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 5683 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 5684 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 5685 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 5686* 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 5687* 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 5688 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 5689 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 5690 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 5691 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 5692 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 )
 ) )
 
Theoremfliftrel 5693*  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 5694* 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 5695* 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 5696* 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 >. ) )
 
Theoremfliftfun 5697* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (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 )   &    |-  ( x  =  y  ->  A  =  C )   &    |-  ( x  =  y  ->  B  =  D )   =>    |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( A  =  C  ->  B  =  D ) ) )
 
Theoremfliftfund 5698* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (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 )   &    |-  ( x  =  y  ->  A  =  C )   &    |-  ( x  =  y  ->  B  =  D )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  A  =  C ) )  ->  B  =  D )   =>    |-  ( ph  ->  Fun  F )
 
Theoremfliftfuns 5699* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (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  ->  ( Fun  F  <->  A. y  e.  X  A. z  e.  X  (
 [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  [_ y  /  x ]_ B  =  [_ z  /  x ]_ B ) ) )
 
Theoremfliftf 5700* The domain and range of the function  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  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  A ) --> S ) )
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