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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdf1st2 6901* An alternate possible definition of the  1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
 
Theoremdf2nd2 6902* An alternate possible definition of the  2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
 
Theorem1stconst 6903 The mapping of a restriction of the  1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
 |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B }
 ) ) : ( A  X.  { B } ) -1-1-onto-> A )
 
Theorem2ndconst 6904 The mapping of a restriction of the  2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
 |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -1-1-onto-> B )
 
Theoremdfmpt2 6905* Alternate definition for the "maps to" notation df-mpt2 6313 (although it requires that  C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  C  e.  _V   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  = 
 U_ x  e.  A  U_ y  e.  B  { <.
 <. x ,  y >. ,  C >. }
 
Theoremmpt2sn 6906* An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)
 |-  F  =  ( x  e.  { A } ,  y  e.  { B }  |->  C )   &    |-  ( x  =  A  ->  C  =  D )   &    |-  (
 y  =  B  ->  D  =  E )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  E  e.  U )  ->  F  =  { <. <. A ,  B >. ,  E >. } )
 
Theoremcurry1 6907* Composition with  `' ( 2nd  |`  ( { C }  X.  _V ) ) turns any binary operation  F with a constant first operand into a function  G of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
 
Theoremcurry1val 6908 The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A )  ->  ( G `  D )  =  ( C F D ) )
 
Theoremcurry1f 6909 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )   =>    |-  ( ( F :
 ( A  X.  B )
 --> D  /\  C  e.  A )  ->  G : B
 --> D )
 
Theoremcurry2 6910* Composition with  `' ( 1st  |`  ( _V  X.  { C }
) ) turns any binary operation  F with a constant second operand into a function  G of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
 
Theoremcurry2f 6911 Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F :
 ( A  X.  B )
 --> D  /\  C  e.  B )  ->  G : A
 --> D )
 
Theoremcurry2val 6912 The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  B )  ->  ( G `  D )  =  ( D F C ) )
 
Theoremcnvf1olem 6913 Lemma for cnvf1o 6914. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) ) 
 ->  ( C  e.  `' A  /\  B  =  U. `' { C } )
 )
 
Theoremcnvf1o 6914* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( Rel  A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
 
Theoremfparlem1 6915 Lemma for fpar 6919. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  =  ( { x }  X.  _V )
 
Theoremfparlem2 6916 Lemma for fpar 6919. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
 y } )  =  ( _V  X.  {
 y } )
 
Theoremfparlem3 6917* Lemma for fpar 6919. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( F  Fn  A  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V 
 X.  _V ) ) ) )  =  U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `
  x ) }  X.  _V ) ) )
 
Theoremfparlem4 6918* Lemma for fpar 6919. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( G  Fn  B  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V 
 X.  _V ) ) ) )  =  U_ y  e.  B  ( ( _V 
 X.  { y } )  X.  ( _V  X.  {
 ( G `  y
 ) } ) ) )
 
Theoremfpar 6919* Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as  z  =  ( ( sqr `  x
)  +  ( abs `  y ) ). (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  H  =  ( ( `' ( 1st  |`  ( _V 
 X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V 
 X.  _V ) ) ) ) )   =>    |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  H  =  ( x  e.  A ,  y  e.  B  |->  <. ( F `
  x ) ,  ( G `  y
 ) >. ) )
 
Theoremfsplit 6920 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 6919 in order to build compound functions such as  y  =  ( ( sqr `  x
)  +  ( abs `  x ) ). (Contributed by NM, 17-Sep-2007.)
 |-  `' ( 1st  |`  _I  )  =  ( x  e.  _V  |->  <. x ,  x >. )
 
Theoremf2ndf 6921 The  2nd (second member of an ordered pair) function restricted to a function  F is a function of  F into the codomain of  F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
 |-  ( F : A --> B  ->  ( 2nd  |`  F ) : F --> B )
 
Theoremfo2ndf 6922 The  2nd (second member of an ordered pair) function restricted to a function  F is a function of  F onto the range of  F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
 |-  ( F : A --> B  ->  ( 2nd  |`  F ) : F -onto-> ran  F )
 
Theoremf1o2ndf1 6923 The  2nd (second member of an ordered pair) function restricted to a one-to-one function  F is a one-to-one function of  F onto the range of  F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
 |-  ( F : A -1-1-> B 
 ->  ( 2nd  |`  F ) : F -1-1-onto-> ran  F )
 
Theoremalgrflem 6924 Lemma for algrf 14611 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( B ( F  o.  1st ) C )  =  ( F `
  B )
 
Theoremfrxp 6925* A lexicographical ordering of two well-founded classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) (Proof shortened by Wolf Lammen, 4-Oct-2014.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  (
 ( R  Fr  A  /\  S  Fr  B ) 
 ->  T  Fr  ( A  X.  B ) )
 
Theoremxporderlem 6926* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  ( <. a ,  b >. T
 <. c ,  d >.  <->  (
 ( ( a  e.  A  /\  c  e.  A )  /\  (
 b  e.  B  /\  d  e.  B )
 )  /\  ( a R c  \/  (
 a  =  c  /\  b S d ) ) ) )
 
Theorempoxp 6927* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  (
 ( R  Po  A  /\  S  Po  B ) 
 ->  T  Po  ( A  X.  B ) )
 
Theoremsoxp 6928* A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  (
 ( R  Or  A  /\  S  Or  B ) 
 ->  T  Or  ( A  X.  B ) )
 
Theoremwexp 6929* A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  ( A  X.  B )  /\  y  e.  ( A  X.  B ) )  /\  ( ( 1st `  x ) R ( 1st `  y
 )  \/  ( ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x ) S ( 2nd `  y )
 ) ) ) }   =>    |-  (
 ( R  We  A  /\  S  We  B ) 
 ->  T  We  ( A  X.  B ) )
 
Theoremfnwelem 6930* Lemma for fnwe 6931. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  A )  /\  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x S y ) ) ) }   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  R  We  B )   &    |-  ( ph  ->  S  We  A )   &    |-  ( ph  ->  ( F " w )  e.  _V )   &    |-  Q  =  { <. u ,  v >.  |  ( ( u  e.  ( B  X.  A )  /\  v  e.  ( B  X.  A ) )  /\  ( ( 1st `  u ) R ( 1st `  v
 )  \/  ( ( 1st `  u )  =  ( 1st `  v
 )  /\  ( 2nd `  u ) S ( 2nd `  v )
 ) ) ) }   &    |-  G  =  ( z  e.  A  |->  <.
 ( F `  z
 ) ,  z >. )   =>    |-  ( ph  ->  T  We  A )
 
Theoremfnwe 6931* A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  A )  /\  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x S y ) ) ) }   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  R  We  B )   &    |-  ( ph  ->  S  We  A )   &    |-  ( ph  ->  ( F " w )  e.  _V )   =>    |-  ( ph  ->  T  We  A )
 
Theoremfnse 6932* Condition for the well-order in fnwe 6931 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  T  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  A )  /\  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x S y ) ) ) }   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  R Se  B )   &    |-  ( ph  ->  ( `' F " w )  e. 
 _V )   =>    |-  ( ph  ->  T Se  A )
 
2.4.8  The support of functions

In this section, the support of functions is defined and corresponding theorems are provided. Since basic properties (see suppval 6935) are based on the Axiom of Union (usage of dmexg 6743), these definition and theorems cannot be provided earlier. Until April 2019, the support of a function was represented by the expression  ( `' R " ( _V  \  { Z } ) ) (see suppimacnv 6944). The theorems which are based on this representation and which are provided in previous sections could be moved into this section to have all related theorems in one section, although they do not depend on the Axiom of Union. This was possible because they are not used before. The current theorems differ from the original ones by requiring that the classes representing the "function" (or its "domain") and the "zero element" are sets. Actually, this does not cause any problem (until now).

 
Syntaxcsupp 6933 Extend class definition to include the support of functions.
 class supp
 
Definitiondf-supp 6934* Define the support of a function against a "zero" value. According to Wikipedia ("Support (mathematics)", 31-Mar-2019, https://en.wikipedia.org/wiki/Support_(mathematics)) "In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero." and "The notion of support also extends in a natural way to functions taking values in more general sets than R [the real numbers] and to other objects.". The following definition allows for such extensions, being applicable for any sets (which usually are functions) and any element (even not necessarily from the range of the function) regarded as "zero". (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
 |- supp  =  ( x  e.  _V ,  z  e.  _V  |->  { i  e.  dom  x  |  ( x " {
 i } )  =/=  { z } }
 )
 
Theoremsuppval 6935* The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
 |-  ( ( X  e.  V  /\  Z  e.  W )  ->  ( X supp  Z )  =  { i  e.  dom  X  |  ( X " { i } )  =/=  { Z } } )
 
Theoremsupp0prc 6936 The support of a class is empty if either the class or the "zero" is a proper class. . (Contributed by AV, 28-May-2019.)
 |-  ( -.  ( X  e.  _V  /\  Z  e.  _V )  ->  ( X supp  Z )  =  (/) )
 
Theoremsuppvalbr 6937* The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
 |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R supp  Z )  =  { x  |  ( E. y  x R y  /\  E. y ( x R y  <->  y  =/=  Z ) ) } )
 
Theoremsupp0 6938 The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
 |-  ( Z  e.  W  ->  ( (/) supp  Z )  =  (/) )
 
Theoremsuppval1 6939* The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
 |-  ( ( Fun  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( X supp  Z )  =  { i  e. 
 dom  X  |  ( X `  i )  =/= 
 Z } )
 
Theoremsuppvalfn 6940* The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
 |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W ) 
 ->  ( F supp  Z )  =  { i  e.  X  |  ( F `
  i )  =/= 
 Z } )
 
Theoremelsuppfn 6941 An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
 |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W ) 
 ->  ( S  e.  ( F supp  Z )  <->  ( S  e.  X  /\  ( F `  S )  =/=  Z ) ) )
 
Theoremcnvimadfsn 6942* The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
 |-  ( `' R "
 ( _V  \  { Z } ) )  =  { x  |  E. y ( x R y  /\  y  =/= 
 Z ) }
 
Theoremsuppimacnvss 6943 The support of functions "defined" by inverse images is a subset of the support defined by df-supp 6934. (Contributed by AV, 7-Apr-2019.)
 |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( `' R " ( _V  \  { Z } ) )  C_  ( R supp  Z ) )
 
Theoremsuppimacnv 6944 Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
 |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R supp  Z )  =  ( `' R " ( _V  \  { Z } ) ) )
 
Theoremfrnsuppeq 6945 Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
 |-  ( ( I  e.  V  /\  Z  e.  W )  ->  ( F : I --> S  ->  ( F supp  Z )  =  ( `' F "
 ( S  \  { Z } ) ) ) )
 
Theoremsuppssdm 6946 The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.)
 |-  ( F supp  Z ) 
 C_  dom  F
 
Theoremsuppsnop 6947 The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
 |-  F  =  { <. X ,  Y >. }   =>    |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F supp 
 Z )  =  if ( Y  =  Z ,  (/) ,  { X } ) )
 
Theoremsnopsuppss 6948 The support of a singleton containing an ordered pair is a subset of the singleton containing the first element of the ordered pair, i.e. it is empty or the singleton itself. (Contributed by AV, 19-Jul-2019.)
 |-  ( { <. X ,  Y >. } supp  Z )  C_ 
 { X }
 
Theoremfvn0elsupp 6949 If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)
 |-  ( ( ( B  e.  V  /\  X  e.  B )  /\  ( G  Fn  B  /\  ( G `  X )  =/=  (/) ) )  ->  X  e.  ( G supp  (/) ) )
 
Theoremfvn0elsuppOLD 6950 Obsolete version of fvn0elsupp 6949 as of 4-Apr-2020. (Contributed by AV, 2-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( ( B  e.  V  /\  X  e.  B )  /\  ( G : B --> A  /\  ( G `  X )  =/=  (/) ) )  ->  X  e.  ( G supp  (/) ) )
 
Theoremfvn0elsuppb 6951 The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)
 |-  ( ( B  e.  V  /\  X  e.  B  /\  G  Fn  B ) 
 ->  ( ( G `  X )  =/=  (/)  <->  X  e.  ( G supp 
 (/) ) ) )
 
Theoremrexsupp 6952* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)
 |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W ) 
 ->  ( E. x  e.  ( F supp  Z )
 ph 
 <-> 
 E. x  e.  X  ( ( F `  x )  =/=  Z  /\  ph ) ) )
 
Theoremressuppss 6953 The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
 |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( ( F  |`  B ) supp  Z ) 
 C_  ( F supp  Z ) )
 
Theoremsuppun 6954 The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.)
 |-  ( ph  ->  G  e.  V )   =>    |-  ( ph  ->  ( F supp  Z )  C_  (
 ( F  u.  G ) supp  Z ) )
 
Theoremressuppssdif 6955 The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
 |-  ( ( F  e.  V  /\  Z  e.  W )  ->  ( F supp  Z )  C_  ( ( ( F  |`  B ) supp  Z )  u.  ( dom 
 F  \  B )
 ) )
 
Theoremmptsuppdifd 6956* The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Z  e.  W )   =>    |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  e.  ( _V  \  { Z }
 ) } )
 
Theoremmptsuppd 6957* The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Z  e.  W )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  U )   =>    |-  ( ph  ->  ( F supp  Z )  =  { x  e.  A  |  B  =/=  Z } )
 
Theoremextmptsuppeq 6958* The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 30-Jun-2019.)
 |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  A 
 C_  B )   &    |-  (
 ( ph  /\  n  e.  ( B  \  A ) )  ->  X  =  Z )   =>    |-  ( ph  ->  (
 ( n  e.  A  |->  X ) supp  Z )  =  ( ( n  e.  B  |->  X ) supp  Z ) )
 
Theoremsuppfnss 6959* The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.)
 |-  ( ( ( F  Fn  A  /\  G  Fn  B )  /\  ( A  C_  B  /\  B  e.  V  /\  Z  e.  W ) )  ->  ( A. x  e.  A  ( ( G `  x )  =  Z  ->  ( F `  x )  =  Z )  ->  ( F supp  Z ) 
 C_  ( G supp  Z ) ) )
 
Theoremfunsssuppss 6960 The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
 |-  ( ( Fun  G  /\  F  C_  G  /\  G  e.  V )  ->  ( F supp  Z ) 
 C_  ( G supp  Z ) )
 
Theoremfnsuppres 6961 Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.)
 |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V )  /\  ( A  i^i  B )  =  (/) )  ->  ( ( F supp  Z )  C_  A  <->  ( F  |`  B )  =  ( B  X.  { Z } ) ) )
 
Theoremfnsuppeq0 6962 The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
 |-  ( ( F  Fn  A  /\  A  e.  W  /\  Z  e.  V ) 
 ->  ( ( F supp  Z )  =  (/)  <->  F  =  ( A  X.  { Z }
 ) ) )
 
Theoremfczsupp0 6963 The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
 |-  ( ( B  X.  { Z } ) supp  Z )  =  (/)
 
Theoremsuppss 6964* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
 |-  ( ph  ->  F : A --> B )   &    |-  (
 ( ph  /\  k  e.  ( A  \  W ) )  ->  ( F `
  k )  =  Z )   =>    |-  ( ph  ->  ( F supp  Z )  C_  W )
 
Theoremsuppssr 6965 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( F supp  Z )  C_  W )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Z  e.  U )   =>    |-  ( ( ph  /\  X  e.  ( A  \  W ) )  ->  ( F `
  X )  =  Z )
 
Theoremsuppssov1 6966* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
 |-  ( ph  ->  (
 ( x  e.  D  |->  A ) supp  Y )  C_  L )   &    |-  ( ( ph  /\  v  e.  R ) 
 ->  ( Y O v )  =  Z )   &    |-  ( ( ph  /\  x  e.  D )  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  D ) 
 ->  B  e.  R )   &    |-  ( ph  ->  Y  e.  W )   =>    |-  ( ph  ->  (
 ( x  e.  D  |->  ( A O B ) ) supp  Z )  C_  L )
 
Theoremsuppssof1 6967* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
 |-  ( ph  ->  ( A supp  Y )  C_  L )   &    |-  ( ( ph  /\  v  e.  R )  ->  ( Y O v )  =  Z )   &    |-  ( ph  ->  A : D --> V )   &    |-  ( ph  ->  B : D
 --> R )   &    |-  ( ph  ->  D  e.  W )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  (
 ( A  oF O B ) supp  Z ) 
 C_  L )
 
Theoremsuppss2 6968* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
 |-  ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( ( k  e.  A  |->  B ) supp  Z )  C_  W )
 
Theoremsuppsssn 6969* Show that the support of a function is a subset of a singleton. (Contributed by AV, 21-Jul-2019.)
 |-  ( ( ph  /\  k  e.  A  /\  k  =/= 
 W )  ->  B  =  Z )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( ( k  e.  A  |->  B ) supp  Z )  C_  { W } )
 
Theoremsuppssfv 6970* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
 |-  ( ph  ->  (
 ( x  e.  D  |->  A ) supp  Y )  C_  L )   &    |-  ( ph  ->  ( F `  Y )  =  Z )   &    |-  (
 ( ph  /\  x  e.  D )  ->  A  e.  V )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( ( x  e.  D  |->  ( F `  A ) ) supp  Z )  C_  L )
 
Theoremsuppofss1d 6971* Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ( ph  /\  x  e.  B )  ->  ( Z X x )  =  Z )   =>    |-  ( ph  ->  (
 ( F  oF X G ) supp  Z ) 
 C_  ( F supp  Z ) )
 
Theoremsuppofss2d 6972* Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ( ph  /\  x  e.  B )  ->  ( x X Z )  =  Z )   =>    |-  ( ph  ->  (
 ( F  oF X G ) supp  Z ) 
 C_  ( G supp  Z ) )
 
Theoremsupp0cosupp0 6973 The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F supp 
 Z )  =  (/)  ->  ( ( F  o.  G ) supp  Z )  =  (/) ) )
 
Theoremimacosupp 6974 The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( Fun 
 G  /\  ( F supp  Z )  C_  ran  G ) 
 ->  ( G " (
 ( F  o.  G ) supp  Z ) )  =  ( F supp  Z ) ) )
 
2.4.9  Special "Maps to" operations

The following theorems are about maps-to operations (see df-mpt2 6313) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 6388, ovmpt2x 6444 and fmpt2x 6878). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short.

 
Theoremopeliunxp2f 6975* Membership in a union of Cartesian products, using bound-variable hypothesis for  E instead of distinct variable conditions as in opeliunxp2 4978. (Contributed by AV, 25-Oct-2020.)
 |-  F/_ x E   &    |-  ( x  =  C  ->  B  =  E )   =>    |-  ( <. C ,  D >.  e.  U_ x  e.  A  ( { x }  X.  B )  <->  ( C  e.  A  /\  D  e.  E ) )
 
Theoremmpt2xeldm 6976* If there is an element of the value of an operation given by a maps-to rule, then the first argument is an element of the first component of the domain and the second argument is an element of the second component of the domain depending on the first argument. (Contributed by AV, 25-Oct-2020.)
 |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( N  e.  ( X F Y ) 
 ->  ( X  e.  C  /\  Y  e.  [_ X  /  x ]_ D ) )
 
Theoremmpt2xneldm 6977* If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.)
 |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( ( X 
 e/  C  \/  Y  e/  [_ X  /  x ]_ D )  ->  ( X F Y )  =  (/) )
 
Theoremmpt2xopn0yelv 6978* If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  ( <. V ,  W >. F K )  ->  K  e.  V ) )
 
Theoremmpt2xopynvov0g 6979* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e/  V ) 
 ->  ( <. V ,  W >. F K )  =  (/) )
 
Theoremmpt2xopxnop0 6980* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( -.  V  e.  ( _V  X.  _V )  ->  ( V F K )  =  (/) )
 
Theoremmpt2xopx0ov0 6981* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( (/) F K )  =  (/)
 
Theoremmpt2xopxprcov0 6982* If the components of the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( -.  ( V  e.  _V  /\  W  e.  _V )  ->  ( <. V ,  W >. F K )  =  (/) )
 
Theoremmpt2xopynvov0 6983* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( K  e/  V  ->  ( <. V ,  W >. F K )  =  (/) )
 
Theoremmpt2xopoveq 6984* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  | 
 ph } )   =>    |-  ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V ) 
 ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph } )
 
Theoremmpt2xopovel 6985* Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  | 
 ph } )   =>    |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  ( <. V ,  W >. F K ) 
 <->  ( K  e.  V  /\  N  e.  V  /\  [.
 <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
 ) )
 
Theoremmpt2xopoveqd 6986* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  | 
 ph } )   &    |-  ( ps  ->  ( V  e.  X  /\  W  e.  Y ) )   &    |-  ( ( ps 
 /\  -.  K  e.  V )  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  =  (/) )   =>    |-  ( ps  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [.
 <. V ,  W >.  /  x ]. [. K  /  y ]. ph } )
 
Theorembrovex 6987* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  C )   &    |-  (
 ( V  e.  _V  /\  E  e.  _V )  ->  Rel  ( V O E ) )   =>    |-  ( F ( V O E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
 
Theorembrovmpt2ex 6988* A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { <. z ,  w >.  |  ph } )   =>    |-  ( F ( V O E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
 ) )
 
Theoremsprmpt2d 6989* The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.)
 |-  M  =  ( v  e.  _V ,  e  e.  _V  |->  { <. x ,  y >.  |  ( x ( v R e ) y  /\  ch ) } )   &    |-  ( ( ph  /\  v  =  V  /\  e  =  E )  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  ( V  e.  _V 
 /\  E  e.  _V ) )   &    |-  ( ph  ->  A. x A. y ( x ( V R E ) y  ->  th ) )   &    |-  ( ph  ->  {
 <. x ,  y >.  | 
 th }  e.  _V )   =>    |-  ( ph  ->  ( V M E )  =  { <. x ,  y >.  |  ( x ( V R E ) y  /\  ps ) } )
 
2.4.10  Function transposition
 
Syntaxctpos 6990 The transposition of a function.
 class tpos  F
 
Definitiondf-tpos 6991* Define the transposition of a function, which is a function  G  = tpos  F satisfying  G ( x ,  y )  =  F ( y ,  x ). (Contributed by Mario Carneiro, 10-Sep-2015.)
 |- tpos  F  =  ( F  o.  ( x  e.  ( `' dom  F  u.  { (/)
 } )  |->  U. `' { x } ) )
 
Theoremtposss 6992 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  C_  G  -> tpos 
 F  C_ tpos  G )
 
Theoremtposeq 6993 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  =  G  -> tpos 
 F  = tpos  G )
 
Theoremtposeqd 6994 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  -> tpos  F  = tpos  G )
 
Theoremtposssxp 6995 The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |- tpos  F  C_  ( ( `'
 dom  F  u.  { (/) } )  X.  ran  F )
 
Theoremreltpos 6996 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |- 
 Rel tpos  F
 
Theorembrtpos2 6997 Value of the transposition at a pair  <. A ,  B >.. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( B  e.  V  ->  ( Atpos  F B  <->  ( A  e.  ( `'
 dom  F  u.  { (/) } )  /\  U. `' { A } F B ) ) )
 
Theorembrtpos0 6998 The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows us to eliminate sethood hypotheses on  A ,  B in brtpos 7000. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  (/) F A ) )
 
Theoremreldmtpos 6999 Necessary and sufficient condition for  dom tpos  F to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom tpos  F  <->  -.  (/)  e.  dom  F )
 
Theorembrtpos 7000 The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( C  e.  V  ->  ( <. A ,  B >.tpos  F C  <->  <. B ,  A >. F C ) )
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