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Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoffres 6001 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( ( F  e.  V  /\  G  e.  W )  ->  ( ( F  oF R G )  |`  D )  =  ( ( F  |`  D )  oF R ( G  |`  D )
 ) )
 
Theoremofmres 6002* Equivalent expressions for a restriction of the function operation map. Unlike  oF R which is a proper class,  (  oF R  |`  ( A  X.  B
) ) can be a set by ofmresex 6003, allowing it to be used as a function or structure argument. By ofmresval 5961, the restricted operation map values are the same as the original values, allowing theorems for  oF R to be reused. (Contributed by NM, 20-Oct-2014.)
 |-  (  oF R  |`  ( A  X.  B ) )  =  (
 f  e.  A ,  g  e.  B  |->  ( f  oF R g ) )
 
Theoremofmresex 6003 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  (  oF R  |`  ( A  X.  B ) )  e.  _V )
 
2.6.14  First and second members of an ordered pair
 
Syntaxc1st 6004 Extend the definition of a class to include the first member an ordered pair function.
 class  1st
 
Syntaxc2nd 6005 Extend the definition of a class to include the second member an ordered pair function.
 class  2nd
 
Definitiondf-1st 6006 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6012 proves that it does this. For example, ( 1st `  <. 3 , 4  >.) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 4990 and op1stb 4369). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 1st  =  ( x  e.  _V  |->  U. dom  { x } )
 
Definitiondf-2nd 6007 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6013 proves that it does this. For example,  ( 2nd ` 
<. 3 , 4 
>.) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 4993 and op2ndb 4992). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
 |- 
 2nd  =  ( x  e.  _V  |->  U. ran  { x } )
 
Theorem1stvalg 6008 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  _V  ->  ( 1st `  A )  =  U. dom  { A } )
 
Theorem2ndvalg 6009 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  e.  _V  ->  ( 2nd `  A )  =  U. ran  { A } )
 
Theorem1st0 6010 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 1st `  (/) )  =  (/)
 
Theorem2nd0 6011 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
 |-  ( 2nd `  (/) )  =  (/)
 
Theoremop1st 6012 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 1st `  <. A ,  B >. )  =  A
 
Theoremop2nd 6013 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( 2nd `  <. A ,  B >. )  =  B
 
Theoremop1std 6014 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( C  =  <. A ,  B >.  ->  ( 1st `  C )  =  A )
 
Theoremop2ndd 6015 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( C  =  <. A ,  B >.  ->  ( 2nd `  C )  =  B )
 
Theoremop1stg 6016 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
 
Theoremop2ndg 6017 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
 
Theoremot1stg 6018 Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6018, ot2ndg 6019, ot3rdgg 6020.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
 
Theoremot2ndg 6019 Extract the second member of an ordered triple. (See ot1stg 6018 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
 
Theoremot3rdgg 6020 Extract the third member of an ordered triple. (See ot1stg 6018 comment.) (Contributed by NM, 3-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
 
Theorem1stval2 6021 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
 
Theorem2ndval2 6022 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
 |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^| |^| `' { A } )
 
Theoremfo1st 6023 The  1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 1st : _V -onto-> _V
 
Theoremfo2nd 6024 The  2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |- 
 2nd : _V -onto-> _V
 
Theoremf1stres 6025 Mapping of a restriction of the 
1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) --> A
 
Theoremf2ndres 6026 Mapping of a restriction of the 
2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) --> B
 
Theoremfo1stresm 6027* Onto mapping of a restriction of the  1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
 |-  ( E. y  y  e.  B  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
 
Theoremfo2ndresm 6028* Onto mapping of a restriction of the  2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
 |-  ( E. x  x  e.  A  ->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> B )
 
Theorem1stcof 6029 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
 |-  ( F : A --> ( B  X.  C ) 
 ->  ( 1st  o.  F ) : A --> B )
 
Theorem2ndcof 6030 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)
 |-  ( F : A --> ( B  X.  C ) 
 ->  ( 2nd  o.  F ) : A --> C )
 
Theoremxp1st 6031 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  ( B  X.  C )  ->  ( 1st `  A )  e.  B )
 
Theoremxp2nd 6032 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  ( B  X.  C )  ->  ( 2nd `  A )  e.  C )
 
Theorem1stexg 6033 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
 |-  ( A  e.  V  ->  ( 1st `  A )  e.  _V )
 
Theorem2ndexg 6034 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
 |-  ( A  e.  V  ->  ( 2nd `  A )  e.  _V )
 
Theoremelxp6 6035 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4996. (Contributed by NM, 9-Oct-2004.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >.  /\  (
 ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
 
Theoremelxp7 6036 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4996. (Contributed by NM, 19-Aug-2006.)
 |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
 
Theoremoprssdmm 6037* Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
 |-  ( ( ph  /\  u  e.  S )  ->  E. v  v  e.  u )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  ( ph  ->  Rel  F )   =>    |-  ( ph  ->  ( S  X.  S )  C_  dom  F )
 
Theoremeqopi 6038 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
 |-  ( ( A  e.  ( V  X.  W ) 
 /\  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C )
 )  ->  A  =  <. B ,  C >. )
 
Theoremxp2 6039* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  X.  B )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x )  e.  B ) }
 
Theoremunielxp 6040 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)
 |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
 
Theorem1st2nd2 6041 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)
 |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theoremxpopth 6042 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)
 |-  ( ( A  e.  ( C  X.  D ) 
 /\  B  e.  ( R  X.  S ) ) 
 ->  ( ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B ) )  <->  A  =  B ) )
 
Theoremeqop 6043 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C )
 ) )
 
Theoremeqop2 6044 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( A  =  <. B ,  C >.  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) ) )
 
Theoremop1steq 6045* Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
 |-  ( A  e.  ( V  X.  W )  ->  ( ( 1st `  A )  =  B  <->  E. x  A  =  <. B ,  x >. ) )
 
Theorem2nd1st 6046 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
 |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A ) >. )
 
Theorem1st2nd 6047 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
 |-  ( ( Rel  B  /\  A  e.  B ) 
 ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A ) >. )
 
Theorem1stdm 6048 The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  ( 1st `  A )  e.  dom  R )
 
Theorem2ndrn 6049 The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( Rel  R  /\  A  e.  R ) 
 ->  ( 2nd `  A )  e.  ran  R )
 
Theorem1st2ndbr 6050 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
 |-  ( ( Rel  B  /\  A  e.  B ) 
 ->  ( 1st `  A ) B ( 2nd `  A ) )
 
Theoremreleldm2 6051* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)
 |-  ( Rel  A  ->  ( B  e.  dom  A  <->  E. x  e.  A  ( 1st `  x )  =  B ) )
 
Theoremreldm 6052* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)
 |-  ( Rel  A  ->  dom 
 A  =  ran  ( x  e.  A  |->  ( 1st `  x ) ) )
 
Theoremsbcopeq1a 6053 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2891 that avoids the existential quantifiers of copsexg 4136). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  <. x ,  y >.  ->  ( [. ( 1st `  A )  /  x ]. [. ( 2nd `  A )  /  y ]. ph  <->  ph ) )
 
Theoremcsbopeq1a 6054 Equality theorem for substitution of a class  A for an ordered pair  <. x ,  y >. in  B (analog of csbeq1a 2983). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  <. x ,  y >.  ->  [_ ( 1st `  A )  /  x ]_ [_ ( 2nd `  A )  /  y ]_ B  =  B )
 
Theoremdfopab2 6055* A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. x ,  y >.  |  ph }  =  { z  e.  ( _V  X.  _V )  | 
 [. ( 1st `  z
 )  /  x ]. [. ( 2nd `  z )  /  y ]. ph }
 
Theoremdfoprab3s 6056* A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  [. ( 1st `  w )  /  x ]. [. ( 2nd `  w )  /  y ]. ph ) }
 
Theoremdfoprab3 6057* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
 |-  ( w  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ps }
 
Theoremdfoprab4 6058* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( w  =  <. x ,  y >.  ->  ( ph 
 <->  ps ) )   =>    |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
 
Theoremdfoprab4f 6059* Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |- 
 F/ x ph   &    |-  F/ y ph   &    |-  ( w  =  <. x ,  y >.  ->  ( ph  <->  ps ) )   =>    |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
 
Theoremdfxp3 6060* Define the cross product of three classes. Compare df-xp 4515. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
 |-  ( ( A  X.  B )  X.  C )  =  { <. <. x ,  y >. ,  z >.  |  ( x  e.  A  /\  y  e.  B  /\  z  e.  C ) }
 
Theoremelopabi 6061* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
 |-  ( x  =  ( 1st `  A )  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  ( 2nd `  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( A  e.  {
 <. x ,  y >.  | 
 ph }  ->  ch )
 
Theoremeloprabi 6062* A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( x  =  ( 1st `  ( 1st `  A ) )  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  ( 2nd `  ( 1st `  A ) )  ->  ( ps  <->  ch ) )   &    |-  ( z  =  ( 2nd `  A )  ->  ( ch  <->  th ) )   =>    |-  ( A  e.  {
 <. <. x ,  y >. ,  z >.  |  ph } 
 ->  th )
 
Theoremmpomptsx 6063* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
 
Theoremmpompts 6064* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
 |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  ( A  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
 
Theoremdmmpossx 6065* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  dom  F  C_  U_ x  e.  A  ( { x }  X.  B )
 
Theoremfmpox 6066* Functionality, domain and codomain of a class given by the maps-to notation, where  B ( x ) is not constant but depends on  x. (Contributed by NM, 29-Dec-2014.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  <->  F : U_ x  e.  A  ( { x }  X.  B ) --> D )
 
Theoremfmpo 6067* Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  <->  F : ( A  X.  B ) --> D )
 
Theoremfnmpo 6068* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B ) )
 
Theoremmpofvex 6069* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( A. x A. y  C  e.  V  /\  R  e.  W  /\  S  e.  X ) 
 ->  ( R F S )  e.  _V )
 
Theoremfnmpoi 6070* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  C  e.  _V   =>    |-  F  Fn  ( A  X.  B )
 
Theoremdmmpo 6071* Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  C  e.  _V   =>    |- 
 dom  F  =  ( A  X.  B )
 
Theoremmpofvexi 6072* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  C  e.  _V   &    |-  R  e.  _V   &    |-  S  e.  _V   =>    |-  ( R F S )  e.  _V
 
Theoremovmpoelrn 6073* An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
 |-  O  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( A. x  e.  A  A. y  e.  B  C  e.  M  /\  X  e.  A  /\  Y  e.  B )  ->  ( X O Y )  e.  M )
 
Theoremdmmpoga 6074* Domain of an operation given by the maps-to notation, closed form of dmmpo 6071. (Contributed by Alexander van der Vekens, 10-Feb-2019.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  dom  F  =  ( A  X.  B ) )
 
Theoremdmmpog 6075* Domain of an operation given by the maps-to notation, closed form of dmmpo 6071. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( C  e.  V  ->  dom  F  =  ( A  X.  B ) )
 
Theoremmpoexxg 6076* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( A  e.  R  /\  A. x  e.  A  B  e.  S )  ->  F  e.  _V )
 
Theoremmpoexg 6077* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( A  e.  R  /\  B  e.  S )  ->  F  e.  _V )
 
Theoremmpoexga 6078* If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  A ,  y  e.  B  |->  C )  e. 
 _V )
 
Theoremmpoex 6079* If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  e.  _V
 
Theoremfnmpoovd 6080* A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)
 |-  ( ph  ->  M  Fn  ( A  X.  B ) )   &    |-  ( ( i  =  a  /\  j  =  b )  ->  D  =  C )   &    |-  ( ( ph  /\  i  e.  A  /\  j  e.  B )  ->  D  e.  U )   &    |-  ( ( ph  /\  a  e.  A  /\  b  e.  B )  ->  C  e.  V )   =>    |-  ( ph  ->  ( M  =  ( a  e.  A ,  b  e.  B  |->  C )  <->  A. i  e.  A  A. j  e.  B  ( i M j )  =  D ) )
 
Theoremfmpoco 6081* Composition of two functions. Variation of fmptco 5554 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  R  e.  C )   &    |-  ( ph  ->  F  =  ( x  e.  A ,  y  e.  B  |->  R ) )   &    |-  ( ph  ->  G  =  ( z  e.  C  |->  S ) )   &    |-  (
 z  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A ,  y  e.  B  |->  T ) )
 
Theoremoprabco 6082* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  D )   &    |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( H `  C ) )   =>    |-  ( H  Fn  D  ->  G  =  ( H  o.  F ) )
 
Theoremoprab2co 6083* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  e.  R )   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  D  e.  S )   &    |-  F  =  ( x  e.  A ,  y  e.  B  |->  <. C ,  D >. )   &    |-  G  =  ( x  e.  A ,  y  e.  B  |->  ( C M D ) )   =>    |-  ( M  Fn  ( R  X.  S )  ->  G  =  ( M  o.  F ) )
 
Theoremdf1st2 6084* 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 6085* 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 6086 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 6087 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 )
 
Theoremdfmpo 6088* Alternate definition for the maps-to notation df-mpo 5747 (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 >. }
 
Theoremcnvf1olem 6089 Lemma for cnvf1o 6090. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) ) 
 ->  ( C  e.  `' A  /\  B  =  U. `' { C } )
 )
 
Theoremcnvf1o 6090* 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 )
 
Theoremf2ndf 6091 The  2nd (second component of an ordered pair) function restricted to a function  F is a function from  F into the codomain of  F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
 |-  ( F : A --> B  ->  ( 2nd  |`  F ) : F --> B )
 
Theoremfo2ndf 6092 The  2nd (second component of an ordered pair) function restricted to a function  F is a function from  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 6093 The  2nd (second component of an ordered pair) function restricted to a one-to-one function  F is a one-to-one function from  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 6094 Lemma for algrf 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 )
 
Theoremalgrflemg 6095 Lemma for algrf 11653 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.)
 |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( B ( F  o.  1st ) C )  =  ( F `  B ) )
 
Theoremxporderlem 6096* 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 6097* 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 ) )
 
Theoremspc2ed 6098* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
 |- 
 F/ x ch   &    |-  F/ y ch   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  ( ps  <->  ch ) )   =>    |-  ( ( ph  /\  ( A  e.  V  /\  B  e.  W )
 )  ->  ( ch  ->  E. x E. y ps ) )
 
Theoremcnvoprab 6099* The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
 |- 
 F/ x ps   &    |-  F/ y ps   &    |-  ( a  = 
 <. x ,  y >.  ->  ( ps  <->  ph ) )   &    |-  ( ps  ->  a  e.  ( _V  X.  _V ) )   =>    |-  `' { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. z ,  a >.  |  ps }
 
Theoremf1od2 6100* Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  C  e.  W )   &    |-  (
 ( ph  /\  z  e.  D )  ->  ( I  e.  X  /\  J  e.  Y )
 )   &    |-  ( ph  ->  (
 ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) ) ) )   =>    |-  ( ph  ->  F : ( A  X.  B ) -1-1-onto-> D )
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