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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpofvex 6101* 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 6102* 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 6103* 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 6104* 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 6105* 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 6106* Domain of an operation given by the maps-to notation, closed form of dmmpo 6103. (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 6107* Domain of an operation given by the maps-to notation, closed form of dmmpo 6103. 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 6108* 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 6109* 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 6110* 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 6111* 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 6112* 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 6113* Composition of two functions. Variation of fmptco 5586 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 6114* 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 6115* 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 6116* 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 6117* 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 6118 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 6119 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 6120* Alternate definition for the maps-to notation df-mpo 5779 (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 6121 Lemma for cnvf1o 6122. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) ) 
 ->  ( C  e.  `' A  /\  B  =  U. `' { C } )
 )
 
Theoremcnvf1o 6122* 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 6123 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 6124 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 6125 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 6126 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 6127 Lemma for algrf 11733 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 6128* 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 6129* 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 6130* 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 6131* 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 6132* 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 6133* 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 6134* 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 )
 
2.6.15  Special maps-to operations

The following theorems are about maps-to operations (see df-mpo 5779) 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 5849, ovmpox 5899 and fmpox 6098). 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 6135* Membership in a union of Cartesian products, using bound-variable hypothesis for  E instead of distinct variable conditions as in opeliunxp2 4679. (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 6136* 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 6137* 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 6138* 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 6139* 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 6140* 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 ) ) )
 
2.6.16  Function transposition
 
Syntaxctpos 6141 The transposition of a function.
 class tpos  F
 
Definitiondf-tpos 6142* 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 6143 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  C_  G  -> tpos 
 F  C_ tpos  G )
 
Theoremtposeq 6144 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  =  G  -> tpos 
 F  = tpos  G )
 
Theoremtposeqd 6145 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  -> tpos  F  = tpos  G )
 
Theoremtposssxp 6146 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 6147 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |- 
 Rel tpos  F
 
Theorembrtpos2 6148 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 6149 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 ) )
 
Theoremreldmtpos 6150 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 )
 
Theorembrtposg 6151 The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( <. A ,  B >.tpos  F C  <->  <. B ,  A >. F C ) )
 
Theoremottposg 6152 The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( <. A ,  B ,  C >.  e. tpos  F  <->  <. B ,  A ,  C >.  e.  F ) )
 
Theoremdmtpos 6153 The domain of tpos  F when  dom  F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  ->  dom tpos  F  =  `' dom  F )
 
Theoremrntpos 6154 The range of tpos  F when  dom  F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  ->  ran tpos  F  =  ran  F )
 
Theoremtposexg 6155 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  e.  V  -> tpos 
 F  e.  _V )
 
Theoremovtposg 6156 The transposition swaps the arguments in a two-argument function. When  F is a matrix, which is to say a function from ( 1 ... m )  X. ( 1 ... n ) to the reals or some ring, tpos  F is the transposition of  F, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( Atpos  F B )  =  ( B F A ) )
 
Theoremtposfun 6157 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Fun  F  ->  Fun tpos  F )
 
Theoremdftpos2 6158* Alternate definition of tpos when 
F has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  -> tpos 
 F  =  ( F  o.  ( x  e.  `' dom  F  |->  U. `' { x } ) ) )
 
Theoremdftpos3 6159* Alternate definition of tpos when 
F has relational domain. Compare df-cnv 4547. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  -> tpos 
 F  =  { <. <. x ,  y >. ,  z >.  |  <. y ,  x >. F z }
 )
 
Theoremdftpos4 6160* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos  F  =  ( F  o.  ( x  e.  (
 ( _V  X.  _V )  u.  { (/) } )  |-> 
 U. `' { x } ) )
 
Theoremtpostpos 6161 Value of the double transposition for a general class  F. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |- tpos tpos  F  =  ( F  i^i  ( ( ( _V 
 X.  _V )  u.  { (/)
 } )  X.  _V ) )
 
Theoremtpostpos2 6162 Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  ( ( Rel  F  /\  Rel  dom  F )  -> tpos tpos  F  =  F )
 
Theoremtposfn2 6163 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
 
Theoremtposfo2 6164 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -onto-> B  -> tpos 
 F : `' A -onto-> B ) )
 
Theoremtposf2 6165 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
 
Theoremtposf12 6166 Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -1-1-> B  -> tpos 
 F : `' A -1-1-> B ) )
 
Theoremtposf1o2 6167 Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -1-1-onto-> B  -> tpos  F : `' A
 -1-1-onto-> B ) )
 
Theoremtposfo 6168 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( F : ( A  X.  B )
 -onto-> C  -> tpos  F : ( B  X.  A )
 -onto-> C )
 
Theoremtposf 6169 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( F : ( A  X.  B ) --> C  -> tpos  F : ( B  X.  A ) --> C )
 
Theoremtposfn 6170 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A ) )
 
Theoremtpos0 6171 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
 |- tpos  (/) 
 =  (/)
 
Theoremtposco 6172 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos 
 ( F  o.  G )  =  ( F  o. tpos  G )
 
Theoremtpossym 6173* Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  A )  ->  (tpos  F  =  F  <->  A. x  e.  A  A. y  e.  A  ( x F y )  =  ( y F x ) ) )
 
Theoremtposeqi 6174 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  G   =>    |- tpos  F  = tpos  G
 
Theoremtposex 6175 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  e.  _V   =>    |- tpos  F  e.  _V
 
Theoremnftpos 6176 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F/_ x F   =>    |-  F/_ xtpos  F
 
Theoremtposoprab 6177* Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  { <. <. x ,  y >. ,  z >.  |  ph }   =>    |- tpos  F  =  { <.
 <. y ,  x >. ,  z >.  |  ph }
 
Theoremtposmpo 6178* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |- tpos  F  =  (
 y  e.  B ,  x  e.  A  |->  C )
 
2.6.17  Undefined values
 
Theorempwuninel2 6179 The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( U. A  e.  V  ->  -.  ~P U. A  e.  A )
 
Theorem2pwuninelg 6180 The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.)
 |-  ( A  e.  V  ->  -.  ~P ~P U. A  e.  A )
 
2.6.18  Functions on ordinals; strictly monotone ordinal functions
 
Theoremiunon 6181* The indexed union of a set of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  e.  On )
 
Syntaxwsmo 6182 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
 wff  Smo  A
 
Definitiondf-smo 6183* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
 |-  ( Smo  A  <->  ( A : dom  A --> On  /\  Ord  dom  A 
 /\  A. x  e.  dom  A
 A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y ) ) ) )
 
Theoremdfsmo2 6184* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( Smo  F  <->  ( F : dom  F --> On  /\  Ord  dom  F 
 /\  A. x  e.  dom  F
 A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
 
Theoremissmo 6185* Conditions for which  A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
 |-  A : B --> On   &    |-  Ord  B   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  e.  y  ->  ( A `  x )  e.  ( A `  y ) ) )   &    |-  dom 
 A  =  B   =>    |-  Smo  A
 
Theoremissmo2 6186* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( F : A --> B  ->  ( ( B 
 C_  On  /\  Ord  A  /\  A. x  e.  A  A. y  e.  x  ( F `  y )  e.  ( F `  x ) )  ->  Smo  F ) )
 
Theoremsmoeq 6187 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( A  =  B  ->  ( Smo  A  <->  Smo  B ) )
 
Theoremsmodm 6188 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( Smo  A  ->  Ord 
 dom  A )
 
Theoremsmores 6189 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( Smo  A  /\  B  e.  dom  A )  ->  Smo  ( A  |`  B ) )
 
Theoremsmores3 6190 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord 
 B )  ->  Smo  ( A  |`  C ) )
 
Theoremsmores2 6191 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
 |-  ( ( Smo  F  /\  Ord  A )  ->  Smo  ( F  |`  A ) )
 
Theoremsmodm2 6192 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
 
Theoremsmofvon2dm 6193 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( ( Smo  F  /\  B  e.  dom  F )  ->  ( F `  B )  e.  On )
 
Theoremiordsmo 6194 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |- 
 Ord  A   =>    |- 
 Smo  (  _I  |`  A )
 
Theoremsmo0 6195 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 6196 If  B is a strictly monotone ordinal function, and  A is in the domain of  B, then the value of the function at 
A is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B )  ->  ( B `  A )  e.  On )
 
Theoremsmoel 6197 If  x is less than  y then a strictly monotone function's value will be strictly less at  x than at  y. (Contributed by Andrew Salmon, 22-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B  /\  C  e.  A ) 
 ->  ( B `  C )  e.  ( B `  A ) )
 
Theoremsmoiun 6198* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B )  ->  U_ x  e.  A  ( B `  x ) 
 C_  ( B `  A ) )
 
Theoremsmoiso 6199 If  F is an isomorphism from an ordinal  A onto  B, which is a subset of the ordinals, then 
F is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  B  C_  On )  ->  Smo  F )
 
Theoremsmoel2 6200 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( B  e.  A  /\  C  e.  B ) )  ->  ( F `  C )  e.  ( F `  B ) )
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