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Theorem List for Metamath Proof Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvtp1g 5901 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 5902 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 5903 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 5904 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
 |-  ( ( B  e.  D  /\  C  e.  A )  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2g 5905 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 5906 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
 |-  B  e.  _V   =>    |-  ( C  e.  A  ->  ( ( A  X.  { B }
 ) `  C )  =  B )
 
Theoremfconst2 5907 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 5908 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 5909 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 5910 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 5909 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 5911 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 5912 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 5913* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5907. (Contributed by NM, 27-Aug-2004.)
 |-  ( F : A --> { B }  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  =  B ) )
 
Theoremfconst3 5914 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
 |-  ( F : A --> { B }  <->  ( F  Fn  A  /\  A  C_  ( `' F " { B } ) ) )
 
Theoremfconst4 5915 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
 |-  ( F : A --> { B }  <->  ( F  Fn  A  /\  ( `' F " { B } )  =  A ) )
 
Theoremresfunexg 5916 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 5917 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 5916 but requires ax-pow 4337. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A  |`  B )  e.  _V )
 
Theoremcofunexg 5918 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 5919 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 5920 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 5916. See fnexALT 5921 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 5921 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 5489. This version of fnex 5920 uses ax-pow 4337, whereas fnex 5920 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 5922 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 5920. (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 5923* 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 5924* 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 5925* 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 5926 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 5922. (Contributed by NM, 11-Nov-1995.)
 |-  ( dom  F  e.  B  ->  ( Fun  F  ->  ran  F  e.  _V ) )
 
Theoremzfrep6 5927* 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 4290 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 4280. (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 5928 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 5929 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 5930 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 4280. (Contributed by NM, 4-Sep-2004.)
 |-  ( ( F : A -1-1-> B  /\  B  e.  C )  ->  A  e.  _V )
 
Theoremeufnfv 5931* 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 5932 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 5933 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 5934 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 5935 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 5936* 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 5937* 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 5938* TODO: This is the same as issref 5206 (which has a much longer proof). Should we replace issref 5206 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 5939* 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 5940* 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 5941* 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 5942* 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 5924, funex 5922, fnex 5920, resfunexg 5916, and funimaexg 5489. See also abrexex2 5960. (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 5943* 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 5944* 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 5945* 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 5946* 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 5947* 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 5948* 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 5949* 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 5950* 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 5951* 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 5952* 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 5953* 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 5954* 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 5955* The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 5954 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 5956* 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 5957* Membership in the union of the range of a function. See elunirnALT 5959 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 5958* 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 5959* Membership in the union of the range of a function, proved directly. Unlike elunirn 5957, it doesn't appeal to ndmfv 5714 (via funiunfv 5954). (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 5960* Existence of an existentially restricted class abstraction.  ph is normally has free-variable parameters  x and  y. See also abrexex 5942. (Contributed by NM, 12-Sep-2004.)
 |-  A  e.  _V   &    |-  { y  |  ph }  e.  _V   =>    |-  { y  |  E. x  e.  A  ph
 }  e.  _V
 
Theoremabexssex 5961* 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 5962* 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 5963* 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 5964 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 5965* 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 5966* 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 5967 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 5968 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 5969 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 5970 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 5971 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 5972* 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 5973 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 5974 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 5975 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 5976 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 5977 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 5978 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 5979 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 5980 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 5981* 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 5982* 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 5983 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 5984 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 5985 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 5986 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 5987 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 5988 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 5989*  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 5990* 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 5991* 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 5992* 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 5993* 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 5994* 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 5995* 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 5996* 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 ) )
 
Theoremfliftval 5997* The value 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 )   &    |-  ( x  =  Y  ->  A  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   &    |-  ( ph  ->  Fun 
 F )   =>    |-  ( ( ph  /\  Y  e.  X )  ->  ( F `  C )  =  D )
 
Theoremisoeq1 5998 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  G  Isom  R ,  S  ( A ,  B ) ) )
 
Theoremisoeq2 5999 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( R  =  T  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  T ,  S  ( A ,  B ) ) )
 
Theoremisoeq3 6000 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( S  =  T  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  T  ( A ,  B ) ) )
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