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Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2ndconst 6001 The mapping of a restriction of the  2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
 |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -1-1-onto-> B )
 
Theoremdfmpt2 6002* Alternate definition for the maps-to notation df-mpt2 5671 (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 6003 Lemma for cnvf1o 6004. (Contributed by Mario Carneiro, 27-Apr-2014.)
 |-  ( ( Rel  A  /\  ( B  e.  A  /\  C  =  U. `' { B } ) ) 
 ->  ( C  e.  `' A  /\  B  =  U. `' { C } )
 )
 
Theoremcnvf1o 6004* 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 6005 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 6006 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 6007 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 6008 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 6009 Lemma for algrf 11366 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 6010* 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 6011* 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 6012* 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 6013* 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 6014* 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 6015* 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 6016* 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-mpt2 5671) where the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 5740, ovmpt2x 5787 and fmpt2x 5984). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short.

 
Theoremopeliunxp2f 6017* Membership in a union of Cartesian products, using bound-variable hypothesis for  E instead of distinct variable conditions as in opeliunxp2 4589. (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 ) )
 
Theoremmpt2xopn0yelv 6018* 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 ) )
 
Theoremmpt2xopoveq 6019* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  | 
 ph } )   =>    |-  ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V ) 
 ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph } )
 
Theoremmpt2xopovel 6020* 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 )
 ) )
 
Theoremsprmpt2 6021* The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  M  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v W e ) p  /\  ch ) } )   &    |-  ( ( v  =  V  /\  e  =  E )  ->  ( ch 
 <->  ps ) )   &    |-  (
 ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V W E ) p 
 ->  th ) )   &    |-  (
 ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  th }  e.  _V )   =>    |-  ( ( V  e.  _V 
 /\  E  e.  _V )  ->  ( V M E )  =  { <. f ,  p >.  |  ( f ( V W E ) p 
 /\  ps ) } )
 
Theoremisprmpt2 6022* 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 ) ) ) )
 
2.6.16  Function transposition
 
Syntaxctpos 6023 The transposition of a function.
 class tpos  F
 
Definitiondf-tpos 6024* 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 6025 Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  C_  G  -> tpos 
 F  C_ tpos  G )
 
Theoremtposeq 6026 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  =  G  -> tpos 
 F  = tpos  G )
 
Theoremtposeqd 6027 Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  -> tpos  F  = tpos  G )
 
Theoremtposssxp 6028 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 6029 The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |- 
 Rel tpos  F
 
Theorembrtpos2 6030 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 6031 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 6032 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 6033 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 6034 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 6035 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 6036 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 6037 The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( F  e.  V  -> tpos 
 F  e.  _V )
 
Theoremovtposg 6038 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 6039 The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Fun  F  ->  Fun tpos  F )
 
Theoremdftpos2 6040* 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 6041* Alternate definition of tpos when 
F has relational domain. Compare df-cnv 4460. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  ( Rel  dom  F  -> tpos 
 F  =  { <. <. x ,  y >. ,  z >.  |  <. y ,  x >. F z }
 )
 
Theoremdftpos4 6042* Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos  F  =  ( F  o.  ( x  e.  (
 ( _V  X.  _V )  u.  { (/) } )  |-> 
 U. `' { x } ) )
 
Theoremtpostpos 6043 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 6044 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 6045 The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F  Fn  A  -> tpos  F  Fn  `' A ) )
 
Theoremtposfo2 6046 Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A -onto-> B  -> tpos 
 F : `' A -onto-> B ) )
 
Theoremtposf2 6047 The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
 |-  ( Rel  A  ->  ( F : A --> B  -> tpos  F : `' A --> B ) )
 
Theoremtposf12 6048 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 6049 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 6050 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 6051 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 6052 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A ) )
 
Theoremtpos0 6053 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
 |- tpos  (/) 
 =  (/)
 
Theoremtposco 6054 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos 
 ( F  o.  G )  =  ( F  o. tpos  G )
 
Theoremtpossym 6055* 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 6056 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  G   =>    |- tpos  F  = tpos  G
 
Theoremtposex 6057 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  e.  _V   =>    |- tpos  F  e.  _V
 
Theoremnftpos 6058 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F/_ x F   =>    |-  F/_ xtpos  F
 
Theoremtposoprab 6059* 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 }
 
Theoremtposmpt2 6060* 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 6061 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 6062 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 6063* 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 6064 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
 wff  Smo  A
 
Definitiondf-smo 6065* 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 6066* 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 6067* 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 6068* 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 6069 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( A  =  B  ->  ( Smo  A  <->  Smo  B ) )
 
Theoremsmodm 6070 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( Smo  A  ->  Ord 
 dom  A )
 
Theoremsmores 6071 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 6072 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 6073 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 6074 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 6075 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 6076 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 6077 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 6078 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 6079 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 6080* 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 6081 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 6082 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 ) )
 
2.6.19  "Strong" transfinite recursion
 
Syntaxcrecs 6083 Notation for a function defined by strong transfinite recursion.
 class recs ( F )
 
Definitiondf-recs 6084* 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 6149 for more details on why this definition is desirable. Unlike df-irdg 6149 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 6114 and tfri2d 6115 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 6085 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
 
Theoremnfrecs 6086 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F/_ x F   =>    |-  F/_ xrecs ( F )
 
Theoremtfrlem1 6087* 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 6088* 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 6089* 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 6090* 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 6091* 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 6092* 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 6093* 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 6094* 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 6095* 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 6096* 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 6097* 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 6098* 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 6099 Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
 |- 
 Fun recs ( F )
 
Theoremtfr2a 6100 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 ) ) )
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