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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremot1stg 6101 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 6101, ot2ndg 6102, ot3rdgg 6103.) (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 6102 Extract the second member of an ordered triple. (See ot1stg 6101 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 6103 Extract the third member of an ordered triple. (See ot1stg 6101 comment.) (Contributed by NM, 3-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
 
Theorem1stval2 6104 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 6105 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 6106 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 6107 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 6108 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 6109 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 6110* 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 6111* 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 6112 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 6113 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 6114 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 6115 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 6116 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
 |-  ( A  e.  V  ->  ( 1st `  A )  e.  _V )
 
Theorem2ndexg 6117 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
 |-  ( A  e.  V  ->  ( 2nd `  A )  e.  _V )
 
Theoremelxp6 6118 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5074. (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 6119 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5074. (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 6120* 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 6121 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 6122* 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 6123 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 6124 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 6125 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 6126 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 6127 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 6128* 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 6129 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 6130 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 6131 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 6132 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 6133 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 6134* 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 6135* 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 6136 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2946 that avoids the existential quantifiers of copsexg 4205). (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 6137 Equality theorem for substitution of a class  A for an ordered pair  <. x ,  y >. in  B (analog of csbeq1a 3040). (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 6138* 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 6139* 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 6140* 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 6141* 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 6142* 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 6143* Define the cross product of three classes. Compare df-xp 4593. (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 6144* 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 6145* 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 6146* 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 6147* 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 6148* 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 6149* 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 6150* 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 6151* 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 6152* 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 6153* 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 6154* 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 6155* 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 6156* 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 6157* Domain of an operation given by the maps-to notation, closed form of dmmpo 6154. (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 6158* Domain of an operation given by the maps-to notation, closed form of dmmpo 6154. 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 6159* 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 6160* 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 6161* 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 6162* 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 6163* 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 6164* Composition of two functions. Variation of fmptco 5634 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 6165* 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 6166* 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 6167* 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 6168* 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 6169 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 6170 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 6171* Alternate definition for the maps-to notation df-mpo 5830 (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 6172 Lemma for cnvf1o 6173. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) ) 
 ->  ( C  e.  `' A  /\  B  =  U. `' { C } )
 )
 
Theoremcnvf1o 6173* 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 6174 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 6175 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 6176 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 6177 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 6178 Lemma for algrf 11926 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 6179* 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 6180* 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 6181* 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 6182* 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 6183* 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 )
 
Theoremdisjxp1 6184* The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  -> Disj  x  e.  A  B )   =>    |-  ( ph  -> Disj  x  e.  A  ( B  X.  C ) )
 
Theoremdisjsnxp 6185* The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |- Disj  j  e.  A  ( {
 j }  X.  B )
 
2.6.16  Special maps-to operations

The following theorems are about maps-to operations (see df-mpo 5830) 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 "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 5900, ovmpox 5950 and fmpox 6149). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short.

 
Theoremopeliunxp2f 6186* Membership in a union of Cartesian products, using bound-variable hypothesis for  E instead of distinct variable conditions as in opeliunxp2 4727. (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 ) )
 
Theoremmpoxopn0yelv 6187* 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 ) )
 
Theoremmpoxopoveq 6188* 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 } )
 
Theoremmpoxopovel 6189* 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 )
 ) )
 
Theoremrbropapd 6190* Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )   &    |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( F  e.  X  /\  P  e.  Y ) 
 ->  ( F M P  <->  ( F W P  /\  ch ) ) ) )
 
Theoremrbropap 6191* Properties of a pair in a restricted binary relation  M expressed as an ordered-pair class abstraction:  M is the binary relation  W restricted by the condition 
ps. (Contributed by AV, 31-Jan-2021.)
 |-  ( ph  ->  M  =  { <. f ,  p >.  |  ( f W p  /\  ps ) } )   &    |-  ( ( f  =  F  /\  p  =  P )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  /\  F  e.  X  /\  P  e.  Y )  ->  ( F M P  <->  ( F W P  /\  ch ) ) )
 
2.6.17  Function transposition
 
Syntaxctpos 6192 The transposition of a function.
 class tpos  F
 
Definitiondf-tpos 6193* 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 6194 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  C_  G  -> tpos 
 F  C_ tpos  G )
 
Theoremtposeq 6195 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  =  G  -> tpos 
 F  = tpos  G )
 
Theoremtposeqd 6196 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  -> tpos  F  = tpos  G )
 
Theoremtposssxp 6197 The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |- tpos  F  C_  ( ( `'
 dom  F  u.  { (/) } )  X.  ran  F )
 
Theoremreltpos 6198 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |- 
 Rel tpos  F
 
Theorembrtpos2 6199 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 6200 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). (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  (/) F A ) )
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