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Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremiunonOLD 6601* 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 6602* 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 6603* 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 6604* A variant of onfununi 6603 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 6605* A variant of onovuni 6604 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 6606* There are no length  om decreasing sequences in the ordinals. See also noinfep 7614 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 6607 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
 wff  Smo  A
 
Definitiondf-smo 6608* 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 6609* 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 6610* 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 6611* Alternative 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 6612 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( A  =  B  ->  ( Smo  A  <->  Smo  B ) )
 
Theoremsmodm 6613 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( Smo  A  ->  Ord 
 dom  A )
 
Theoremsmores 6614 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 6615 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 6616 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 6617 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 6618 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 6619 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 6620 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 6621 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 6622 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 6623* 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 6624 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 6625 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 6626 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 6627 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 6628 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 6629 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 6630 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 6631 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 ) ) )
 
2.4.22  "Strong" transfinite recursion
 
Syntaxcrecs 6632 Notation for a function defined by strong transfinite recursion.
 class recs ( F )
 
Definitiondf-recs 6633* 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 6668 for more details on why this definition is desirable. Unlike df-rdg 6668 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 6661 and recsval 6662 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7501, zorn2 8386, and dfac8alem 7910. (Contributed by Stefan O'Rear, 18-Jan-2015.) (New usage is discouraged.)

 |- recs
 ( F )  = 
 U. { f  | 
 E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }
 
Theoremrecseq 6634 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
 
Theoremnfrecs 6635 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F/_ x F   =>    |-  F/_ xrecs ( F )
 
Theoremtfrlem1 6636* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( A  e.  On  ->  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A. x  e.  A  ( ( F `
  x )  =  ( B `  ( F  |`  x ) ) 
 /\  ( G `  x )  =  ( B `  ( G  |`  x ) ) )  ->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
 
Theoremtfrlem2 6637* Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6636 into the main proof. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G )  ->  ( A  e.  On  ->  ( A. w ( A  e.  On  ->  ( w  e.  A  ->  ( ( F `  w )  =  ( B `  ( F  |`  w ) )  /\  ( G `
  w )  =  ( B `  ( G  |`  w ) ) ) ) )  ->  y  =  z )
 ) ) )
 
Theoremtfrlem3 6638* 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. y  e.  z  ( g `  y
 )  =  ( F `
  ( g  |`  y ) ) ) }
 
Theoremtfrlem3a 6639* 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, 22-Jul-2012.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  A  =  { g  |  E. x  e.  On  ( g  Fn  x  /\  A. y  e.  x  ( g `  y
 )  =  ( F `
  ( g  |`  y ) ) ) }
 
Theoremtfrlem4 6640* 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 6641* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (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  /\  h  e.  A )  ->  (
 ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h )  ->  u  =  v ) )
 
Theoremrecsfval 6642* 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 6643* 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 6644* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Fun recs ( F )
 
Theoremtfrlem8 6645* 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 6646* 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 6647* Lemma for transfinite recursion. Without using ax-rep 4320, 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 6648* 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 6652, 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 6649* 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 6650* 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 6651* 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 6652* Lemma for transfinite recursion. Assuming ax-rep 4320,  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 6653* Lemma for transfinite recursion. Without assuming ax-rep 4320, 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 6654* Lemma for finite recursion. Without assuming ax-rep 4320, 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 6655 A weak version of tfr1 6658 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 6656 A weak version of tfr2 6659 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 6657 Without assuming ax-rep 4320, 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 6658 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 6659 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 6660* 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 )
 
Theoremrecsfnon 6661 Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |- recs
 ( F )  Fn 
 On
 
Theoremrecsval 6662 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 6663*  G is a function. Lemma for tz7.44-1 6664, tz7.44-2 6665, and tz7.44-3 6666. (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 6664* 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 6665* 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 6666* 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 ) )
 
2.4.23  Recursive definition generator
 
Syntaxcrdg 6667 Extend class notation with the recursive definition generator, with characteristic function  F and initial value  I.
 class  rec ( F ,  I
 )
 
Definitiondf-rdg 6668* 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 6658 and  G in tz7.44-1 6664 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. But once we get past this hurdle, otherwise recursive definitions become relatively simple, as in for example oav 6755, from which we prove the recursive textbook definition as theorems oa0 6760, oasuc 6768, and oalim 6776 (with the help of theorems rdg0 6679, rdgsuc 6682, and rdglim2a 6691). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6693 and frsuc 6694. 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 3740) 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 11324 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 11567 and integer powers df-exp 11383.

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 6669 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 6670 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 6671 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 6672 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 6673* 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 6674 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |- 
 Fun  rec ( F ,  A )
 
Theoremrdgdmlim 6675 The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)
 |- 
 Lim  dom  rec ( F ,  A )
 
Theoremrdgfnon 6676 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 6677* 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 6678* 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 6679 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 6680 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 6681 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 6682 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 6683 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 6684 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 6685 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
 |-  ( A  e.  C  ->  ( rec ( F ,  A ) `  (/) )  =  A )
 
Theoremrdgsucmptf 6686 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 6687 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 6686 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 6688* This version of rdgsucmpt 6689 avoids the not-free hypothesis of rdgsucmptf 6686 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 6689* 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 6690* 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 6691* 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 ) )
 
2.4.24  Finite recursion
 
Theoremfrfnom 6692 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
 
Theoremfr0g 6693 The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.)
 |-  ( A  e.  B  ->  ( ( rec ( F ,  A )  |` 
 om ) `  (/) )  =  A )
 
Theoremfrsuc 6694 The successor value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( B  e.  om  ->  ( ( rec ( F ,  A )  |` 
 om ) `  suc  B )  =  ( F `
  ( ( rec ( F ,  A )  |`  om ) `  B ) ) )
 
Theoremfrsucmpt 6695 The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x D   &    |-  F  =  ( rec ( ( x  e. 
 _V  |->  C ) ,  A )  |`  om )   &    |-  ( x  =  ( F `  B )  ->  C  =  D )   =>    |-  ( ( B  e.  om 
 /\  D  e.  V )  ->  ( F `  suc  B )  =  D )
 
Theoremfrsucmptn 6696 The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 
D is a proper class). This is a technical lemma that can be used together with frsucmpt 6695 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x D   &    |-  F  =  ( rec ( ( x  e. 
 _V  |->  C ) ,  A )  |`  om )   &    |-  ( x  =  ( F `  B )  ->  C  =  D )   =>    |-  ( -.  D  e.  _V 
 ->  ( F `  suc  B )  =  (/) )
 
Theoremfrsucmpt2 6697* The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  C ) ,  A )  |`  om )   &    |-  (
 y  =  x  ->  E  =  C )   &    |-  (
 y  =  ( F `
  B )  ->  E  =  D )   =>    |-  (
 ( B  e.  om  /\  D  e.  V ) 
 ->  ( F `  suc  B )  =  D )
 
Theoremtz7.48lem 6698* A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.)
 |-  F  Fn  On   =>    |-  ( ( A 
 C_  On  /\  A. x  e.  A  A. y  e.  x  -.  ( F `
  x )  =  ( F `  y
 ) )  ->  Fun  `' ( F  |`  A ) )
 
Theoremtz7.48-2 6699* Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)
 |-  F  Fn  On   =>    |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F )
 
Theoremtz7.48-1 6700* Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
 |-  F  Fn  On   =>    |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A )
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