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Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwfr2a 7001 A weak version of wfr2 7003 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( X  e.  dom  F 
 ->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X ) ) ) )
 
Theoremwfr1 7002 The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function  G. Then, using a base class  A and a well-ordering  R of  A, we define a function  F. This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of  F. We begin by showing that  F is a function over  A. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  F  Fn  A
 
Theoremwfr2 7003 The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of  F at any  X  e.  A is  G recursively applied to all "previous" values of  F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( X  e.  A  ->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X ) ) ) )
 
Theoremwfr3 7004* The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that  F is unique. We do this by showing that any function  H with the same properties we proved of  F in wfr1 7002 and wfr2 7003 is identical to  F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( ( H  Fn  A  /\  A. z  e.  A  ( H `  z )  =  ( G `  ( H  |`  Pred ( R ,  A ,  z ) ) ) )  ->  F  =  H )
 
2.4.14  Functions on ordinals; strictly monotone ordinal functions
 
Theoremiunon 7005* 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 )
 
TheoremiunonOLD 7006* 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.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  e.  On  -> 
 U_ x  e.  A  B  e.  On )
 
Theoremiinon 7007* The nonempty indexed intersection of a class of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_
 x  e.  A  B  e.  On )
 
Theoremonfununi 7008* A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  ( Lim  y  ->  ( F `  y )  =  U_ x  e.  y  ( F `  x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( F `  x ) 
 C_  ( F `  y ) )   =>    |-  ( ( S  e.  T  /\  S  C_ 
 On  /\  S  =/=  (/) )  ->  ( F ` 
 U. S )  = 
 U_ x  e.  S  ( F `  x ) )
 
Theoremonovuni 7009* A variant of onfununi 7008 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  ( Lim  y  ->  ( A F y )  =  U_ x  e.  y  ( A F x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x ) 
 C_  ( A F y ) )   =>    |-  ( ( S  e.  T  /\  S  C_ 
 On  /\  S  =/=  (/) )  ->  ( A F U. S )  = 
 U_ x  e.  S  ( A F x ) )
 
Theoremonoviun 7010* A variant of onovuni 7009 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( Lim  y  ->  ( A F y )  =  U_ x  e.  y  ( A F x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x ) 
 C_  ( A F y ) )   =>    |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( A F U_ z  e.  K  L )  =  U_ z  e.  K  ( A F L ) )
 
Theoremonnseq 7011* There are no length  om decreasing sequences in the ordinals. See also noinfep 8110 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  ( ( F `  (/) )  e.  On  ->  E. x  e.  om  -.  ( F `  suc  x )  e.  ( F `  x ) )
 
Syntaxwsmo 7012 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
 wff  Smo  A
 
Definitiondf-smo 7013* 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 7014* 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 7015* 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 7016* 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 7017 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( A  =  B  ->  ( Smo  A  <->  Smo  B ) )
 
Theoremsmodm 7018 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( Smo  A  ->  Ord 
 dom  A )
 
Theoremsmores 7019 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 7020 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 7021 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 7022 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 )
 
Theoremsmofvon2 7023 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( Smo  F  ->  ( F `  B )  e.  On )
 
Theoremiordsmo 7024 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 7025 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 7026 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 7027 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 7028* 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 7029 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 7030 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 ) )
 
Theoremsmo11 7031 A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( F : A
 --> B  /\  Smo  F )  ->  F : A -1-1-> B )
 
Theoremsmoord 7032 A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
 
Theoremsmoword 7033 A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  C_  D  <->  ( F `  C )  C_  ( F `
  D ) ) )
 
Theoremsmogt 7034 A strictly monotone ordinal function is greater than or equal to its argument. Exercise 1 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 23-Nov-2011.) (Revised by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( F  Fn  A  /\  Smo  F  /\  C  e.  A )  ->  C  C_  ( F `  C ) )
 
Theoremsmorndom 7035 The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)
 |-  ( ( F : A
 --> B  /\  Smo  F  /\  Ord  B )  ->  A  C_  B )
 
Theoremsmoiso2 7036 The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of  On. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( ( Ord  A  /\  B  C_  On )  ->  ( ( F : A -onto-> B  /\  Smo  F ) 
 <->  F  Isom  _E  ,  _E  ( A ,  B ) ) )
 
2.4.15  "Strong" transfinite recursion
 
Syntaxcrecs 7037 Notation for a function defined by strong transfinite recursion.
 class recs ( F )
 
Definitiondf-recs 7038 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-rdg 7076 for more details on why this definition is desirable. Unlike df-rdg 7076 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 recsfnon 7069 and recsval 7070 for the primary contract of this definition. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Scott Fenton, 3-Aug-2020.)
 |- recs
 ( F )  = wrecs
 (  _E  ,  On ,  F )
 
Theoremdfrecs3 7039* The old definition of transfinite recursion. This version is preferred for developement, as it demonstrates the properties of transfinite recursion without relying on well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.)
 |- recs
 ( F )  = 
 U. { f  | 
 E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }
 
Theoremrecseq 7040 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
 
Theoremnfrecs 7041 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F/_ x F   =>    |-  F/_ xrecs ( F )
 
Theoremtfrlem1 7042* 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 ) )
 
Theoremtfrlem3a 7043* 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 7044* 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 ) ) ) }
 
Theoremtfrlem4 7045* 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 7046* 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 7047* 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 7048* 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 7049* 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 7050* 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 7051* 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 ) ) )
 
Theoremtfrlem9a 7052* Lemma for transfinite recursion. Without using ax-rep 4472, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  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 )  e.  _V )
 
Theoremtfrlem10 7053* Lemma for transfinite recursion. We define class  C by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to,  On. Using this assumption we will prove facts about  C that will lead to a contradiction in tfrlem14 7057, thus showing the domain of recs does in fact equal  On. Here we show (under the false assumption) that  C is a function extending the domain of recs
( F ) by one. (Contributed by NM, 14-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 ) ) ) }   &    |-  C  =  (recs ( F )  u. 
 { <. dom recs ( F ) ,  ( F ` recs
 ( F ) )
 >. } )   =>    |-  ( dom recs ( F )  e.  On  ->  C  Fn  suc  dom recs ( F ) )
 
Theoremtfrlem11 7054* Lemma for transfinite recursion. Compute the value of  C. (Contributed by NM, 18-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 ) ) ) }   &    |-  C  =  (recs ( F )  u. 
 { <. dom recs ( F ) ,  ( F ` recs
 ( F ) )
 >. } )   =>    |-  ( dom recs ( F )  e.  On  ->  ( B  e.  suc  dom recs ( F )  ->  ( C `  B )  =  ( F `  ( C  |`  B ) ) ) )
 
Theoremtfrlem12 7055* Lemma for transfinite recursion. Show  C is an acceptable function. (Contributed by NM, 15-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 ) ) ) }   &    |-  C  =  (recs ( F )  u. 
 { <. dom recs ( F ) ,  ( F ` recs
 ( F ) )
 >. } )   =>    |-  (recs ( F )  e.  _V  ->  C  e.  A )
 
Theoremtfrlem13 7056* Lemma for transfinite recursion. If recs is a set function, then  C is acceptable, and thus a subset of recs, but 
dom  C is bigger than  dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 -. recs ( F )  e. 
 _V
 
Theoremtfrlem14 7057* Lemma for transfinite recursion. Assuming ax-rep 4472,  dom recs  e.  _V  <-> recs  e. 
_V, so since  dom recs is an ordinal, it must be equal to  On. (Contributed by NM, 14-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 ) ) ) }   =>    |- 
 dom recs ( F )  =  On
 
Theoremtfrlem15 7058* Lemma for transfinite recursion. Without assuming ax-rep 4472, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 14-Nov-2014.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( B  e.  On  ->  ( B  e.  dom recs ( F )  <->  (recs ( F )  |`  B )  e.  _V ) )
 
Theoremtfrlem16 7059* Lemma for finite recursion. Without assuming ax-rep 4472, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Lim  dom recs ( F )
 
Theoremtfr1a 7060 A weak version of tfr1 7063 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   =>    |-  ( Fun  F  /\  Lim 
 dom  F )
 
Theoremtfr2a 7061 A weak version of tfr2 7064 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   =>    |-  ( A  e.  dom  F 
 ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
 
Theoremtfr2b 7062 Without assuming ax-rep 4472, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   =>    |-  ( Ord  A  ->  ( A  e.  dom  F  <->  ( F  |`  A )  e.  _V ) )
 
Theoremtfr1 7063 Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class  G, normally a function, and define a class  A of all "acceptable" functions. The final function we're interested in is the union  F  = recs ( G ) of them.  F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of  F. In this first part we show that  F is a function whose domain is all ordinal numbers. (Contributed by NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
 |-  F  = recs ( G )   =>    |-  F  Fn  On
 
Theoremtfr2 7064 Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function  F has the property that for any function  G whatsoever, the "next" value of  F is  G recursively applied to all "previous" values of  F. (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( G )   =>    |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
 
Theoremtfr3 7065* Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. Finally, we show that  F is unique. We do this by showing that any class  B with the same properties of  F that we showed in parts 1 and 2 is identical to  F. (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( G )   =>    |-  ( ( B  Fn  On  /\  A. x  e. 
 On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
 
Theoremtfr1ALT 7066 Alternate proof of tfr1 7063 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  F  = recs ( G )   =>    |-  F  Fn  On
 
Theoremtfr2ALT 7067 Alternate proof of tfr2 7064 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  F  = recs ( G )   =>    |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
 
Theoremtfr3ALT 7068* Alternate proof of tfr3 7065 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  F  = recs ( G )   =>    |-  ( ( B  Fn  On  /\  A. x  e. 
 On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
 
Theoremrecsfnon 7069 Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |- recs
 ( F )  Fn 
 On
 
Theoremrecsval 7070 Strong transfinite recursion in terms of all previous values. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( A  e.  On  ->  (recs ( F ) `
  A )  =  ( F `  (recs ( F )  |`  A ) ) )
 
Theoremtz7.44lem1 7071*  G is a function. Lemma for tz7.44-1 7072, tz7.44-2 7073, and tz7.44-3 7074. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  G  =  { <. x ,  y >.  |  ( ( x  =  (/)  /\  y  =  A )  \/  ( -.  ( x  =  (/)  \/  Lim  dom 
 x )  /\  y  =  ( H `  ( x `  U. dom  x ) ) )  \/  ( Lim  dom  x  /\  y  =  U. ran  x ) ) }   =>    |-  Fun  G
 
Theoremtz7.44-1 7072* The value of  F at  (/). Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  G  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `  U. dom  x ) ) ) ) )   &    |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )   &    |-  A  e.  _V   =>    |-  ( (/)  e.  X  ->  ( F `  (/) )  =  A )
 
Theoremtz7.44-2 7073* The value of  F at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  G  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `  U. dom  x ) ) ) ) )   &    |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )   &    |-  ( y  e.  X  ->  ( F  |`  y )  e.  _V )   &    |-  F  Fn  X   &    |-  Ord  X   =>    |-  ( suc  B  e.  X  ->  ( F `  suc  B )  =  ( H `  ( F `
  B ) ) )
 
Theoremtz7.44-3 7074* The value of  F at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  G  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `  U. dom  x ) ) ) ) )   &    |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )   &    |-  ( y  e.  X  ->  ( F  |`  y )  e.  _V )   &    |-  F  Fn  X   &    |-  Ord  X   =>    |-  ( ( B  e.  X  /\  Lim  B )  ->  ( F `  B )  =  U. ( F
 " B ) )
 
2.4.16  Recursive definition generator
 
Syntaxcrdg 7075 Extend class notation with the recursive definition generator, with characteristic function  F and initial value  I.
 class  rec ( F ,  I
 )
 
Definitiondf-rdg 7076* Define a recursive definition generator on  On (the class of ordinal numbers) with characteristic function  F and initial value  I. This combines functions  F in tfr1 7063 and  G in tz7.44-1 7072 into one definition. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our  rec operation (especially when df-recs 7038 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple, as in for example oav 7161, from which we prove the recursive textbook definition as theorems oa0 7166, oasuc 7174, and oalim 7182 (with the help of theorems rdg0 7087, rdgsuc 7090, and rdglim2a 7099). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 7101 and frsuc 7102. Our  rec operation apparently does not appear in published literature, although closely related is Definition 25.2 of [Quine] p. 177, which he uses to "turn...a recursion into a genuine or direct definition" (p. 174). Note that the  if operations (see df-if 3848) select cases based on whether the domain of  g is zero, a successor, or a limit ordinal.

An important use of this definition is in the recursive sequence generator df-seq 12157 on the natural numbers (as a subset of the complex numbers), allowing us to define, with direct definitions, recursive infinite sequences such as the factorial function df-fac 12403 and integer powers df-exp 12216.

Note: We introduce 
rec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

 |- 
 rec ( F ,  I )  = recs (
 ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `
  ( g `  U.
 dom  g ) ) ) ) ) )
 
Theoremrdgeq1 7077 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( F  =  G  ->  rec ( F ,  A )  =  rec ( G ,  A ) )
 
Theoremrdgeq2 7078 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( A  =  B  ->  rec ( F ,  A )  =  rec ( F ,  B ) )
 
Theoremrdgeq12 7079 Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.)
 |-  ( ( F  =  G  /\  A  =  B )  ->  rec ( F ,  A )  =  rec ( G ,  B ) )
 
Theoremnfrdg 7080 Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ x rec ( F ,  A )
 
Theoremrdglem1 7081* Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use. (Contributed by NM, 9-Apr-1995.)
 |- 
 { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }  =  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) }
 
Theoremrdgfun 7082 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |- 
 Fun  rec ( F ,  A )
 
Theoremrdgdmlim 7083 The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)
 |- 
 Lim  dom  rec ( F ,  A )
 
Theoremrdgfnon 7084 The recursive definition generator is a function on ordinal numbers. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
 |- 
 rec ( F ,  A )  Fn  On
 
Theoremrdgvalg 7085* Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  B )  =  ( ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `
  ( g `  U.
 dom  g ) ) ) ) ) `  ( rec ( F ,  A )  |`  B ) ) )
 
Theoremrdgval 7086* Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( B  e.  On  ->  ( rec ( F ,  A ) `  B )  =  (
 ( g  e.  _V  |->  if ( g  =  (/) ,  A ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `
  ( g `  U.
 dom  g ) ) ) ) ) `  ( rec ( F ,  A )  |`  B ) ) )
 
Theoremrdg0 7087 The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  A  e.  _V   =>    |-  ( rec ( F ,  A ) `  (/) )  =  A
 
Theoremrdgseg 7088 The initial segments of the recursive definition generator are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A )  |`  B )  e.  _V )
 
Theoremrdgsucg 7089 The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.)
 |-  ( B  e.  dom  rec ( F ,  A )  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A ) `  B ) ) )
 
Theoremrdgsuc 7090 The value of the recursive definition generator at a successor. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( B  e.  On  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `  ( rec ( F ,  A ) `  B ) ) )
 
Theoremrdglimg 7091 The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)
 |-  ( ( B  e.  dom 
 rec ( F ,  A )  /\  Lim  B )  ->  ( rec ( F ,  A ) `  B )  =  U. ( rec ( F ,  A ) " B ) )
 
Theoremrdglim 7092 The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A ) `  B )  =  U. ( rec ( F ,  A ) " B ) )
 
Theoremrdg0g 7093 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
 |-  ( A  e.  C  ->  ( rec ( F ,  A ) `  (/) )  =  A )
 
Theoremrdgsucmptf 7094 The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x D   &    |-  F  =  rec (
 ( x  e.  _V  |->  C ) ,  A )   &    |-  ( x  =  ( F `  B ) 
 ->  C  =  D )   =>    |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
 
Theoremrdgsucmptnf 7095 The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class  D is a proper class). This is a technical lemma that can be used together with rdgsucmptf 7094 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x D   &    |-  F  =  rec (
 ( x  e.  _V  |->  C ) ,  A )   &    |-  ( x  =  ( F `  B ) 
 ->  C  =  D )   =>    |-  ( -.  D  e.  _V  ->  ( F `  suc  B )  =  (/) )
 
Theoremrdgsucmpt2 7096* This version of rdgsucmpt 7097 avoids the not-free hypothesis of rdgsucmptf 7094 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  F  =  rec (
 ( x  e.  _V  |->  C ) ,  A )   &    |-  ( y  =  x 
 ->  E  =  C )   &    |-  ( y  =  ( F `  B )  ->  E  =  D )   =>    |-  (
 ( B  e.  On  /\  D  e.  V ) 
 ->  ( F `  suc  B )  =  D )
 
Theoremrdgsucmpt 7097* The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by Mario Carneiro, 9-Sep-2013.)
 |-  F  =  rec (
 ( x  e.  _V  |->  C ) ,  A )   &    |-  ( x  =  ( F `  B ) 
 ->  C  =  D )   =>    |-  ( ( B  e.  On  /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
 
Theoremrdglim2 7098* The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)
 |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A ) `  B )  =  U. { y  |  E. x  e.  B  y  =  ( rec ( F ,  A ) `  x ) } )
 
Theoremrdglim2a 7099* The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998.)
 |-  ( ( B  e.  C  /\  Lim  B )  ->  ( rec ( F ,  A ) `  B )  =  U_ x  e.  B  ( rec ( F ,  A ) `  x ) )
 
2.4.17  Finite recursion
 
Theoremfrfnom 7100 The function generated by finite recursive definition generation is a function on omega. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 14-Nov-2014.)
 |-  ( rec ( F ,  A )  |`  om )  Fn  om
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40127
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