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Theorem List for Intuitionistic Logic Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnvf1o 6201* 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 6202 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 6203 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 6204 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 6205 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 6206 Lemma for algrf 11986 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 6207* 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 6208* 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 6209* 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 6210* 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 6211* 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 6212* 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 6213* 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 5855) 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 5928, ovmpox 5978 and fmpox 6176). 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 6214* Membership in a union of Cartesian products, using bound-variable hypothesis for  E instead of distinct variable conditions as in opeliunxp2 4749. (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 6215* 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 6216* 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 6217* 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 6218* 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 6219* 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 6220 The transposition of a function.
 class tpos  F
 
Definitiondf-tpos 6221* 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 6222 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  C_  G  -> tpos 
 F  C_ tpos  G )
 
Theoremtposeq 6223 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  =  G  -> tpos 
 F  = tpos  G )
 
Theoremtposeqd 6224 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  -> tpos  F  = tpos  G )
 
Theoremtposssxp 6225 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 6226 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |- 
 Rel tpos  F
 
Theorembrtpos2 6227 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 6228 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 6229 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 6230 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 6231 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 6232 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 6233 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 6234 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  e.  V  -> tpos 
 F  e.  _V )
 
Theoremovtposg 6235 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 6236 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Fun  F  ->  Fun tpos  F )
 
Theoremdftpos2 6237* 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 6238* Alternate definition of tpos when 
F has relational domain. Compare df-cnv 4617. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  -> tpos 
 F  =  { <. <. x ,  y >. ,  z >.  |  <. y ,  x >. F z }
 )
 
Theoremdftpos4 6239* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos  F  =  ( F  o.  ( x  e.  (
 ( _V  X.  _V )  u.  { (/) } )  |-> 
 U. `' { x } ) )
 
Theoremtpostpos 6240 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 6241 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 6242 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
 
Theoremtposfo2 6243 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -onto-> B  -> tpos 
 F : `' A -onto-> B ) )
 
Theoremtposf2 6244 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
 
Theoremtposf12 6245 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 6246 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 6247 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 6248 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 6249 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A ) )
 
Theoremtpos0 6250 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
 |- tpos  (/) 
 =  (/)
 
Theoremtposco 6251 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos 
 ( F  o.  G )  =  ( F  o. tpos  G )
 
Theoremtpossym 6252* 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 6253 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  G   =>    |- tpos  F  = tpos  G
 
Theoremtposex 6254 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  e.  _V   =>    |- tpos  F  e.  _V
 
Theoremnftpos 6255 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F/_ x F   =>    |-  F/_ xtpos  F
 
Theoremtposoprab 6256* 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 6257* 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 )
 
2.6.18  Undefined values
 
Theorempwuninel2 6258 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 6259 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 )
 
2.6.19  Functions on ordinals; strictly monotone ordinal functions
 
Theoremiunon 6260* 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 6261 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
 wff  Smo  A
 
Definitiondf-smo 6262* 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 6263* 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 6264* 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 6265* 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 6266 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( A  =  B  ->  ( Smo  A  <->  Smo  B ) )
 
Theoremsmodm 6267 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( Smo  A  ->  Ord 
 dom  A )
 
Theoremsmores 6268 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 6269 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 6270 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 6271 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 6272 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 6273 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 6274 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 6275 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 6276 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 6277* 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 6278 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 6279 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 ) )
 
2.6.20  "Strong" transfinite recursion
 
Syntaxcrecs 6280 Notation for a function defined by strong transfinite recursion.
 class recs ( F )
 
Definitiondf-recs 6281* Define a function recs ( F ) on  On, the class of ordinal numbers, by transfinite recursion given a rule  F which sets the next value given all values so far. See df-irdg 6346 for more details on why this definition is desirable. Unlike df-irdg 6346 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See tfri1d 6311 and tfri2d 6312 for the primary contract of this definition.

(Contributed by Stefan O'Rear, 18-Jan-2015.)

 |- recs
 ( F )  = 
 U. { f  | 
 E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }
 
Theoremrecseq 6282 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
 
Theoremnfrecs 6283 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F/_ x F   =>    |-  F/_ xrecs ( F )
 
Theoremtfrlem1 6284* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  ( Fun  F  /\  A  C_ 
 dom  F ) )   &    |-  ( ph  ->  ( Fun  G  /\  A  C_  dom  G ) )   &    |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( B `  ( F  |`  x ) ) )   &    |-  ( ph  ->  A. x  e.  A  ( G `  x )  =  ( B `  ( G  |`  x ) ) )   =>    |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
 
Theoremtfrlem3ag 6285* Lemma for transfinite recursion. This lemma just changes some bound variables in  A for later use. (Contributed by Jim Kingdon, 5-Jul-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( G  e.  _V  ->  ( G  e.  A  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) ) )
 
Theoremtfrlem3a 6286* Lemma for transfinite recursion. Let  A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in  A for later use. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  G  e.  _V   =>    |-  ( G  e.  A  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
 
Theoremtfrlem3 6287* Lemma for transfinite recursion. Let  A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in  A for later use. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  A  =  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) }
 
Theoremtfrlem3-2d 6288* Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)
 |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e. 
 _V ) )   =>    |-  ( ph  ->  ( Fun  F  /\  ( F `  g )  e. 
 _V ) )
 
Theoremtfrlem4 6289* Lemma for transfinite recursion.  A is the class of all "acceptable" functions, and  F is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( g  e.  A  ->  Fun  g )
 
Theoremtfrlem5 6290* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( ( g  e.  A  /\  h  e.  A )  ->  (
 ( x g u 
 /\  x h v )  ->  u  =  v ) )
 
Theoremrecsfval 6291* Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- recs
 ( F )  = 
 U. A
 
Theoremtfrlem6 6292* Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Rel recs ( F )
 
Theoremtfrlem7 6293* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Fun recs ( F )
 
Theoremtfrlem8 6294* Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Ord  dom recs ( F )
 
Theoremtfrlem9 6295* Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( B  e.  dom recs ( F )  ->  (recs ( F ) `  B )  =  ( F `  (recs ( F )  |`  B ) ) )
 
Theoremtfrfun 6296 Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
 |- 
 Fun recs ( F )
 
Theoremtfr2a 6297 A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   =>    |-  ( A  e.  dom  F 
 ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
 
Theoremtfr0dm 6298 Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)
 |-  F  = recs ( G )   =>    |-  ( ( G `  (/) )  e.  V  ->  (/)  e. 
 dom  F )
 
Theoremtfr0 6299 Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
 |-  F  = recs ( G )   =>    |-  ( ( G `  (/) )  e.  V  ->  ( F `  (/) )  =  ( G `  (/) ) )
 
Theoremtfrlemisucfn 6300* We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6308. (Contributed by Jim Kingdon, 2-Jul-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  ( ph  ->  z  e.  On )   &    |-  ( ph  ->  g  Fn  z )   &    |-  ( ph  ->  g  e.  A )   =>    |-  ( ph  ->  (
 g  u.  { <. z ,  ( F `  g ) >. } )  Fn  suc  z )
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