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Theorem List for Intuitionistic Logic Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvmptdf 5501* Alternate deduction version of fvmpt 5491, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  D )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  (
 ( F `  A )  =  B  ->  ps ) )   &    |-  F/_ x F   &    |-  F/ x ps   =>    |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) )
 
Theoremfvmptdv 5502* Alternate deduction version of fvmpt 5491, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  D )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  (
 ( F `  A )  =  B  ->  ps ) )   =>    |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ps ) )
 
Theoremfvmptdv2 5503* Alternate deduction version of fvmpt 5491, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  A  e.  D )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  B  =  C )   =>    |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ( F `  A )  =  C ) )
 
Theoremmpteqb 5504* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5511. (Contributed by Mario Carneiro, 14-Nov-2014.)
 |-  ( A. x  e.  A  B  e.  V  ->  ( ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C )  <->  A. x  e.  A  B  =  C )
 )
 
Theoremfvmptt 5505* Closed theorem form of fvmpt 5491. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  ( ( A. x ( x  =  A  ->  B  =  C ) 
 /\  F  =  ( x  e.  D  |->  B )  /\  ( A  e.  D  /\  C  e.  V ) )  ->  ( F `  A )  =  C )
 
Theoremfvmptf 5506* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5490 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   &    |-  F  =  ( x  e.  D  |->  B )   =>    |-  ( ( A  e.  D  /\  C  e.  V )  ->  ( F `  A )  =  C )
 
Theoremfvmptd3 5507* Deduction version of fvmpt 5491. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
 |-  F  =  ( x  e.  D  |->  B )   &    |-  ( x  =  A  ->  B  =  C )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  C  e.  V )   =>    |-  ( ph  ->  ( F `  A )  =  C )
 
Theoremelfvmptrab1 5508* Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  F  =  ( x  e.  V  |->  { y  e.  [_ x  /  m ]_ M  |  ph } )   &    |-  ( X  e.  V  ->  [_ X  /  m ]_ M  e.  _V )   =>    |-  ( Y  e.  ( F `  X )  ->  ( X  e.  V  /\  Y  e.  [_ X  /  m ]_ M ) )
 
Theoremelfvmptrab 5509* Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  F  =  ( x  e.  V  |->  { y  e.  M  |  ph } )   &    |-  ( X  e.  V  ->  M  e.  _V )   =>    |-  ( Y  e.  ( F `  X ) 
 ->  ( X  e.  V  /\  Y  e.  M ) )
 
Theoremfvopab6 5510* Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  F  =  { <. x ,  y >.  |  (
 ph  /\  y  =  B ) }   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  D  /\  C  e.  R  /\  ps )  ->  ( F `  A )  =  C )
 
Theoremeqfnfv 5511* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G 
 <-> 
 A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
 
Theoremeqfnfv2 5512* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G 
 <->  ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
 
Theoremeqfnfv3 5513* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G 
 <->  ( B  C_  A  /\  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) ) ) )
 
Theoremeqfnfvd 5514* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremeqfnfv2f 5515* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5511 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
 |-  F/_ x F   &    |-  F/_ x G   =>    |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
 
Theoremeqfunfv 5516* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
 |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G 
 /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
 
Theoremfvreseq 5517* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
 |-  ( ( ( F  Fn  A  /\  G  Fn  A )  /\  B  C_  A )  ->  (
 ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
 
Theoremfndmdif 5518* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  { x  e.  A  |  ( F `  x )  =/=  ( G `  x ) } )
 
Theoremfndmdifcom 5519 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  dom  ( G  \  F ) )
 
Theoremfndmin 5520* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
 
Theoremfneqeql 5521 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G 
 <-> 
 dom  ( F  i^i  G )  =  A ) )
 
Theoremfneqeql2 5522 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G 
 <->  A  C_  dom  ( F  i^i  G ) ) )
 
Theoremfnreseql 5523 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  ->  ( ( F  |`  X )  =  ( G  |`  X )  <->  X  C_  dom  ( F  i^i  G ) ) )
 
Theoremchfnrn 5524* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
 |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  x )  ->  ran  F  C_  U. A )
 
Theoremfunfvop 5525 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `  A ) >.  e.  F )
 
Theoremfunfvbrb 5526 Two ways to say that  A is in the domain of  F. (Contributed by Mario Carneiro, 1-May-2014.)
 |-  ( Fun  F  ->  ( A  e.  dom  F  <->  A F ( F `  A ) ) )
 
Theoremfvimacnvi 5527 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
 |-  ( ( Fun  F  /\  A  e.  ( `' F " B ) )  ->  ( F `  A )  e.  B )
 
Theoremfvimacnv 5528 The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5196 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( ( F `
  A )  e.  B  <->  A  e.  ( `' F " B ) ) )
 
Theoremfunimass3 5529 A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5528 would be the special case of  A being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( ( F " A )  C_  B  <->  A  C_  ( `' F " B ) ) )
 
Theoremfunimass5 5530* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( A  C_  ( `' F " B )  <->  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremfunconstss 5531* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( A. x  e.  A  ( F `  x )  =  B  <->  A 
 C_  ( `' F " { B } )
 ) )
 
Theoremelpreima 5532 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( F  Fn  A  ->  ( B  e.  ( `' F " C )  <-> 
 ( B  e.  A  /\  ( F `  B )  e.  C )
 ) )
 
Theoremfniniseg 5533 Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( F  Fn  A  ->  ( C  e.  ( `' F " { B } )  <->  ( C  e.  A  /\  ( F `  C )  =  B ) ) )
 
Theoremfncnvima2 5534* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  ( `' F " B )  =  { x  e.  A  |  ( F `  x )  e.  B } )
 
Theoremfniniseg2 5535* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  ( `' F " { B } )  =  { x  e.  A  |  ( F `  x )  =  B }
 )
 
Theoremfnniniseg2 5536* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( F  Fn  A  ->  ( `' F "
 ( _V  \  { B } ) )  =  { x  e.  A  |  ( F `  x )  =/=  B } )
 
Theoremrexsupp 5537* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  ( F  Fn  A  ->  ( E. x  e.  ( `' F "
 ( _V  \  { Z } ) ) ph  <->  E. x  e.  A  (
 ( F `  x )  =/=  Z  /\  ph )
 ) )
 
Theoremunpreima 5538 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( Fun  F  ->  ( `' F " ( A  u.  B ) )  =  ( ( `' F " A )  u.  ( `' F " B ) ) )
 
Theoreminpreima 5539 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
 |-  ( Fun  F  ->  ( `' F " ( A  i^i  B ) )  =  ( ( `' F " A )  i^i  ( `' F " B ) ) )
 
Theoremdifpreima 5540 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
 |-  ( Fun  F  ->  ( `' F " ( A 
 \  B ) )  =  ( ( `' F " A ) 
 \  ( `' F " B ) ) )
 
Theoremrespreima 5541 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( Fun  F  ->  ( `' ( F  |`  B )
 " A )  =  ( ( `' F " A )  i^i  B ) )
 
Theoremfimacnv 5542 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
 |-  ( F : A --> B  ->  ( `' F " B )  =  A )
 
Theoremfnopfv 5543 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  <. B ,  ( F `  B ) >.  e.  F )
 
Theoremfvelrn 5544 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
 |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  ( F `  A )  e.  ran  F )
 
Theoremfnfvelrn 5545 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F `  B )  e.  ran  F )
 
Theoremffvelrn 5546 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
 |-  ( ( F : A
 --> B  /\  C  e.  A )  ->  ( F `
  C )  e.  B )
 
Theoremffvelrni 5547 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
 |-  F : A --> B   =>    |-  ( C  e.  A  ->  ( F `  C )  e.  B )
 
Theoremffvelrnda 5548 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ( ph  /\  C  e.  A ) 
 ->  ( F `  C )  e.  B )
 
Theoremffvelrnd 5549 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  ( F `  C )  e.  B )
 
Theoremrexrn 5550* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( F  Fn  A  ->  ( E. x  e.  ran  F ph  <->  E. y  e.  A  ps ) )
 
Theoremralrn 5551* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)
 |-  ( x  =  ( F `  y ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( F  Fn  A  ->  ( A. x  e.  ran  F ph  <->  A. y  e.  A  ps ) )
 
Theoremelrnrexdm 5552* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 |-  ( Fun  F  ->  ( Y  e.  ran  F  ->  E. x  e.  dom  F  Y  =  ( F `
  x ) ) )
 
Theoremelrnrexdmb 5553* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  ( Fun  F  ->  ( Y  e.  ran  F  <->  E. x  e.  dom  F  Y  =  ( F `  x ) ) )
 
Theoremeldmrexrn 5554* For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 |-  ( Fun  F  ->  ( Y  e.  dom  F  ->  E. x  e.  ran  F  x  =  ( F `
  Y ) ) )
 
Theoremralrnmpt 5555* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( A. x  e.  A  B  e.  V  ->  ( A. y  e. 
 ran  F ps  <->  A. x  e.  A  ch ) )
 
Theoremrexrnmpt 5556* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( y  =  B  ->  ( ps  <->  ch ) )   =>    |-  ( A. x  e.  A  B  e.  V  ->  ( E. y  e. 
 ran  F ps  <->  E. x  e.  A  ch ) )
 
Theoremdff2 5557 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  F  C_  ( A  X.  B ) ) )
 
Theoremdff3im 5558* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |-  ( F : A --> B  ->  ( F  C_  ( A  X.  B ) 
 /\  A. x  e.  A  E! y  x F y ) )
 
Theoremdff4im 5559* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |-  ( F : A --> B  ->  ( F  C_  ( A  X.  B ) 
 /\  A. x  e.  A  E! y  e.  B  x F y ) )
 
Theoremdffo3 5560* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
 
Theoremdffo4 5561* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
 
Theoremdffo5 5562* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  A. y  e.  B  E. x  x F y ) )
 
Theoremfmpt 5563* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  C )   =>    |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
 
Theoremf1ompt 5564* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
 |-  F  =  ( x  e.  A  |->  C )   =>    |-  ( F : A -1-1-onto-> B  <->  ( A. x  e.  A  C  e.  B  /\  A. y  e.  B  E! x  e.  A  y  =  C )
 )
 
Theoremfmpti 5565* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( x  e.  A  ->  C  e.  B )   =>    |-  F : A --> B
 
Theoremfvmptelrn 5566* The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
 |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )   =>    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  C )
 
Theoremfmptd 5567* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremfmpttd 5568* Version of fmptd 5567 with inlined definition. Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by BJ, 16-Aug-2022.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  C )   =>    |-  ( ph  ->  ( x  e.  A  |->  B ) : A --> C )
 
Theoremfmpt3d 5569* Domain and codomain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  C )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremfmptdf 5570* A version of fmptd 5567 using bound-variable hypothesis instead of a distinct variable condition for  ph. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremffnfv 5571* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremffnfvf 5572 A function maps to a class to which all values belong. This version of ffnfv 5571 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x F   =>    |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
 
Theoremfnfvrnss 5573* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)
 |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
 
Theoremrnmptss 5574* The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  C  ->  ran 
 F  C_  C )
 
Theoremfmpt2d 5575* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ( ph  /\  y  e.  A )  ->  ( F `  y )  e.  C )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremffvresb 5576* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
 |-  ( Fun  F  ->  ( ( F  |`  A ) : A --> B  <->  A. x  e.  A  ( x  e.  dom  F 
 /\  ( F `  x )  e.  B ) ) )
 
Theoremresflem 5577* A lemma to bound the range of a restriction. The conclusion would also hold with  ( X  i^i  Y ) in place of  Y (provided  x does not occur in  X). If that stronger result is needed, it is however simpler to use the instance of resflem 5577 where  ( X  i^i  Y ) is substituted for  Y (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.)
 |-  ( ph  ->  F : V --> X )   &    |-  ( ph  ->  A  C_  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  ( F `  x )  e.  Y )   =>    |-  ( ph  ->  ( F  |`  A ) : A --> Y )
 
Theoremf1oresrab 5578* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
 |-  F  =  ( x  e.  A  |->  C )   &    |-  ( ph  ->  F : A
 -1-1-onto-> B )   &    |-  ( ( ph  /\  x  e.  A  /\  y  =  C )  ->  ( ch  <->  ps ) )   =>    |-  ( ph  ->  ( F  |`  { x  e.  A  |  ps }
 ) : { x  e.  A  |  ps } -1-1-onto-> {
 y  e.  B  |  ch } )
 
Theoremfmptco 5579* Composition of two functions expressed as ordered-pair class abstractions. If  F has the equation ( x + 2 ) and  G the equation ( 3 * z ) then  ( G  o.  F
) has the equation ( 3 * ( x + 2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  B )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )   &    |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )   &    |-  (
 y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A  |->  T ) )
 
Theoremfmptcof 5580* Version of fmptco 5579 where  ph needn't be distinct from  x. (Contributed by NM, 27-Dec-2014.)
 |-  ( ph  ->  A. x  e.  A  R  e.  B )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )   &    |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )   &    |-  ( y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A  |->  T ) )
 
Theoremfmptcos 5581* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ph  ->  A. x  e.  A  R  e.  B )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )   &    |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A  |->  [_ R  /  y ]_ S ) )
 
Theoremcofmpt 5582* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
 |-  ( ph  ->  F : C --> D )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  C )   =>    |-  ( ph  ->  ( F  o.  ( x  e.  A  |->  B ) )  =  ( x  e.  A  |->  ( F `  B ) ) )
 
Theoremfcompt 5583* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
 |-  ( ( A : D
 --> E  /\  B : C
 --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
  x ) ) ) )
 
Theoremfcoconst 5584 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  ( I  X.  { Y } ) )  =  ( I  X.  {
 ( F `  Y ) } ) )
 
Theoremfsn 5585 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } )
 
Theoremfsng 5586 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F : { A } --> { B } 
 <->  F  =  { <. A ,  B >. } )
 )
 
Theoremfsn2 5587 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)
 |-  A  e.  _V   =>    |-  ( F : { A } --> B  <->  ( ( F `
  A )  e.  B  /\  F  =  { <. A ,  ( F `  A ) >. } ) )
 
Theoremxpsng 5588 The cross product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  { B }
 )  =  { <. A ,  B >. } )
 
Theoremxpsn 5589 The cross product of two singletons. (Contributed by NM, 4-Nov-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( { A }  X.  { B } )  =  { <. A ,  B >. }
 
Theoremdfmpt 5590 Alternate definition for the maps-to notation df-mpt 3986 (although it requires that  B be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
 |-  B  e.  _V   =>    |-  ( x  e.  A  |->  B )  = 
 U_ x  e.  A  { <. x ,  B >. }
 
Theoremfnasrn 5591 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   =>    |-  ( x  e.  A  |->  B )  = 
 ran  ( x  e.  A  |->  <. x ,  B >. )
 
Theoremdfmptg 5592 Alternate definition for the maps-to notation df-mpt 3986 (which requires that  B be a set). (Contributed by Jim Kingdon, 9-Jan-2019.)
 |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. } )
 
Theoremfnasrng 5593 A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)
 |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  =  ran  ( x  e.  A  |->  <. x ,  B >. ) )
 
Theoremressnop0 5594 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 5595 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 5596 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 5597 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 5598 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 5599 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
  B ) >. } )
 
Theoremfressnfv 5600 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 ) )
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