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Theorem List for Intuitionistic Logic Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsmo0 6201 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 6202 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 6203 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 6204* 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 6205 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 6206 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 6207 Notation for a function defined by strong transfinite recursion.
 class recs ( F )
 
Definitiondf-recs 6208* 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 6273 for more details on why this definition is desirable. Unlike df-irdg 6273 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 6238 and tfri2d 6239 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 6209 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
 
Theoremnfrecs 6210 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F/_ x F   =>    |-  F/_ xrecs ( F )
 
Theoremtfrlem1 6211* 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 6212* 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 6213* 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 6214* 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 6215* 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 6216* 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 6217* 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 6218* 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 6219* 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 6220* 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 6221* 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 6222* 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 6223 Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
 |- 
 Fun recs ( F )
 
Theoremtfr2a 6224 A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   =>    |-  ( A  e.  dom  F 
 ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
 
Theoremtfr0dm 6225 Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)
 |-  F  = recs ( G )   =>    |-  ( ( G `  (/) )  e.  V  ->  (/)  e. 
 dom  F )
 
Theoremtfr0 6226 Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
 |-  F  = recs ( G )   =>    |-  ( ( G `  (/) )  e.  V  ->  ( F `  (/) )  =  ( G `  (/) ) )
 
Theoremtfrlemisucfn 6227* We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6235. (Contributed by Jim Kingdon, 2-Jul-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  ( ph  ->  z  e.  On )   &    |-  ( ph  ->  g  Fn  z )   &    |-  ( ph  ->  g  e.  A )   =>    |-  ( ph  ->  (
 g  u.  { <. z ,  ( F `  g ) >. } )  Fn  suc  z )
 
Theoremtfrlemisucaccv 6228* We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6235. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  ( ph  ->  z  e.  On )   &    |-  ( ph  ->  g  Fn  z )   &    |-  ( ph  ->  g  e.  A )   =>    |-  ( ph  ->  (
 g  u.  { <. z ,  ( F `  g ) >. } )  e.  A )
 
Theoremtfrlemibacc 6229* Each element of  B is an acceptable function. Lemma for tfrlemi1 6235. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
  g ) >. } ) ) }   &    |-  ( ph  ->  x  e.  On )   &    |-  ( ph  ->  A. z  e.  x  E. g
 ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  B 
 C_  A )
 
Theoremtfrlemibxssdm 6230* The union of  B is defined on all ordinals. Lemma for tfrlemi1 6235. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
  g ) >. } ) ) }   &    |-  ( ph  ->  x  e.  On )   &    |-  ( ph  ->  A. z  e.  x  E. g
 ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  x 
 C_  dom  U. B )
 
Theoremtfrlemibfn 6231* The union of  B is a function defined on  x. Lemma for tfrlemi1 6235. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
  g ) >. } ) ) }   &    |-  ( ph  ->  x  e.  On )   &    |-  ( ph  ->  A. z  e.  x  E. g
 ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  U. B  Fn  x )
 
Theoremtfrlemibex 6232* The set  B exists. Lemma for tfrlemi1 6235. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
  g ) >. } ) ) }   &    |-  ( ph  ->  x  e.  On )   &    |-  ( ph  ->  A. z  e.  x  E. g
 ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  B  e.  _V )
 
Theoremtfrlemiubacc 6233* The union of  B satisfies the recursion rule (lemma for tfrlemi1 6235). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
  g ) >. } ) ) }   &    |-  ( ph  ->  x  e.  On )   &    |-  ( ph  ->  A. z  e.  x  E. g
 ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  A. u  e.  x  (
 U. B `  u )  =  ( F `  ( U. B  |`  u ) ) )
 
Theoremtfrlemiex 6234* Lemma for tfrlemi1 6235. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   &    |-  B  =  { h  |  E. z  e.  x  E. g ( g  Fn  z  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( F `
  g ) >. } ) ) }   &    |-  ( ph  ->  x  e.  On )   &    |-  ( ph  ->  A. z  e.  x  E. g
 ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  E. f ( f  Fn  x  /\  A. u  e.  x  ( f `  u )  =  ( F `  ( f  |`  u ) ) ) )
 
Theoremtfrlemi1 6235* We can define an acceptable function on any ordinal.

As with many of the transfinite recursion theorems, we have a hypothesis that states that  F is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   =>    |-  ( ( ph  /\  C  e.  On )  ->  E. g
 ( g  Fn  C  /\  A. u  e.  C  ( g `  u )  =  ( F `  ( g  |`  u ) ) ) )
 
Theoremtfrlemi14d 6236* The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   =>    |-  ( ph  ->  dom recs ( F )  =  On )
 
Theoremtfrexlem 6237* The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  A. x ( Fun  F  /\  ( F `  x )  e.  _V )
 )   =>    |-  ( ( ph  /\  C  e.  V )  ->  (recs ( F ) `  C )  e.  _V )
 
Theoremtfri1d 6238* Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that  G is defined "everywhere", which is stated here as  ( G `  x )  e.  _V. Alternately,  A. x  e.  On A. f ( f  Fn  x  -> 
f  e.  dom  G
) would suffice.

Given a function  G satisfying that condition, we 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 Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

 |-  F  = recs ( G )   &    |-  ( ph  ->  A. x ( Fun  G  /\  ( G `  x )  e.  _V )
 )   =>    |-  ( ph  ->  F  Fn  On )
 
Theoremtfri2d 6239* Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule  G ( as described at tfri1 6268). Here we show that the function  F has the property that for any function  G satisfying that condition, the "next" value of  F is  G recursively applied to all "previous" values of  F. (Contributed by Jim Kingdon, 4-May-2019.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  A. x ( Fun  G  /\  ( G `  x )  e.  _V )
 )   =>    |-  ( ( ph  /\  A  e.  On )  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
 
Theoremtfr1onlem3ag 6240* Lemma for transfinite recursion. This lemma changes some bound variables in  A (version of tfrlem3ag 6212 but for tfr1on 6253 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.)
 |-  A  =  { f  |  E. x  e.  X  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  y ) ) ) }   =>    |-  ( H  e.  V  ->  ( H  e.  A  <->  E. z  e.  X  ( H  Fn  z  /\  A. w  e.  z  ( H `  w )  =  ( G `  ( H  |`  w ) ) ) ) )
 
Theoremtfr1onlem3 6241* Lemma for transfinite recursion. This lemma changes some bound variables in  A (version of tfrlem3 6214 but for tfr1on 6253 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.)
 |-  A  =  { f  |  E. x  e.  X  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  y ) ) ) }   =>    |-  A  =  { g  |  E. z  e.  X  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) }
 
Theoremtfr1onlemssrecs 6242* Lemma for tfr1on 6253. The union of functions acceptable for tfr1on 6253 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.)
 |-  A  =  { f  |  E. x  e.  X  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  Ord 
 X )   =>    |-  ( ph  ->  U. A  C_ recs
 ( G ) )
 
Theoremtfr1onlemsucfn 6243* We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6253. (Contributed by Jim Kingdon, 12-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  z  e.  X )   &    |-  ( ph  ->  g  Fn  z )   &    |-  ( ph  ->  g  e.  A )   =>    |-  ( ph  ->  ( g  u.  { <. z ,  ( G `  g ) >. } )  Fn  suc  z )
 
Theoremtfr1onlemsucaccv 6244* Lemma for tfr1on 6253. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  z  e.  Y )   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  g  Fn  z )   &    |-  ( ph  ->  g  e.  A )   =>    |-  ( ph  ->  (
 g  u.  { <. z ,  ( G `  g ) >. } )  e.  A )
 
Theoremtfr1onlembacc 6245* Lemma for tfr1on 6253. Each element of  B is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g
 ( g  Fn  z  /\  g  e.  A  /\  h  =  (
 g  u.  { <. z ,  ( G `  g ) >. } )
 ) }   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  B  C_  A )
 
Theoremtfr1onlembxssdm 6246* Lemma for tfr1on 6253. The union of  B is defined on all elements of  X. (Contributed by Jim Kingdon, 14-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g
 ( g  Fn  z  /\  g  e.  A  /\  h  =  (
 g  u.  { <. z ,  ( G `  g ) >. } )
 ) }   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  D  C_ 
 dom  U. B )
 
Theoremtfr1onlembfn 6247* Lemma for tfr1on 6253. The union of  B is a function defined on  x. (Contributed by Jim Kingdon, 15-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g
 ( g  Fn  z  /\  g  e.  A  /\  h  =  (
 g  u.  { <. z ,  ( G `  g ) >. } )
 ) }   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  U. B  Fn  D )
 
Theoremtfr1onlembex 6248* Lemma for tfr1on 6253. The set  B exists. (Contributed by Jim Kingdon, 14-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g
 ( g  Fn  z  /\  g  e.  A  /\  h  =  (
 g  u.  { <. z ,  ( G `  g ) >. } )
 ) }   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  B  e.  _V )
 
Theoremtfr1onlemubacc 6249* Lemma for tfr1on 6253. The union of  B satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g
 ( g  Fn  z  /\  g  e.  A  /\  h  =  (
 g  u.  { <. z ,  ( G `  g ) >. } )
 ) }   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u ) ) )
 
Theoremtfr1onlemex 6250* Lemma for tfr1on 6253. (Contributed by Jim Kingdon, 16-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g
 ( g  Fn  z  /\  g  e.  A  /\  h  =  (
 g  u.  { <. z ,  ( G `  g ) >. } )
 ) }   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )   =>    |-  ( ph  ->  E. f
 ( f  Fn  D  /\  A. u  e.  D  ( f `  u )  =  ( G `  ( f  |`  u ) ) ) )
 
Theoremtfr1onlemaccex 6251* We can define an acceptable function on any element of  X.

As with many of the transfinite recursion theorems, we have hypotheses that state that  F is a function and that it is defined up to  X. (Contributed by Jim Kingdon, 16-Mar-2022.)

 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   =>    |-  ( ( ph  /\  C  e.  X )  ->  E. g
 ( g  Fn  C  /\  A. u  e.  C  ( g `  u )  =  ( G `  ( g  |`  u ) ) ) )
 
Theoremtfr1onlemres 6252* Lemma for tfr1on 6253. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  A  =  { f  |  E. x  e.  X  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  Y  C_ 
 dom  F )
 
Theoremtfr1on 6253* Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f  Fn  x )  ->  ( G `  f )  e.  _V )   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  Y  C_ 
 dom  F )
 
Theoremtfri1dALT 6254* Alternate proof of tfri1d 6238 in terms of tfr1on 6253.

Although this does show that the tfr1on 6253 proof is general enough to also prove tfri1d 6238, the tfri1d 6238 proof is simpler in places because it does not need to deal with 
X being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.)

 |-  F  = recs ( G )   &    |-  ( ph  ->  A. x ( Fun  G  /\  ( G `  x )  e.  _V )
 )   =>    |-  ( ph  ->  F  Fn  On )
 
Theoremtfrcllemssrecs 6255* Lemma for tfrcl 6267. The union of functions acceptable for tfrcl 6267 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.)
 |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  Ord 
 X )   =>    |-  ( ph  ->  U. A  C_ recs
 ( G ) )
 
Theoremtfrcllemsucfn 6256* We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6267. (Contributed by Jim Kingdon, 24-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  z  e.  X )   &    |-  ( ph  ->  g : z --> S )   &    |-  ( ph  ->  g  e.  A )   =>    |-  ( ph  ->  ( g  u.  { <. z ,  ( G `  g ) >. } ) : suc  z --> S )
 
Theoremtfrcllemsucaccv 6257* Lemma for tfrcl 6267. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  z  e.  Y )   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  g :
 z --> S )   &    |-  ( ph  ->  g  e.  A )   =>    |-  ( ph  ->  (
 g  u.  { <. z ,  ( G `  g ) >. } )  e.  A )
 
Theoremtfrcllembacc 6258* Lemma for tfrcl 6267. Each element of  B is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `  g ) >. } ) ) }   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g : z --> S  /\  A. w  e.  z  (
 g `  w )  =  ( G `  (
 g  |`  w ) ) ) )   =>    |-  ( ph  ->  B  C_  A )
 
Theoremtfrcllembxssdm 6259* Lemma for tfrcl 6267. The union of  B is defined on all elements of  X. (Contributed by Jim Kingdon, 25-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `  g ) >. } ) ) }   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g : z --> S  /\  A. w  e.  z  (
 g `  w )  =  ( G `  (
 g  |`  w ) ) ) )   =>    |-  ( ph  ->  D  C_ 
 dom  U. B )
 
Theoremtfrcllembfn 6260* Lemma for tfrcl 6267. The union of  B is a function defined on  x. (Contributed by Jim Kingdon, 25-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `  g ) >. } ) ) }   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g : z --> S  /\  A. w  e.  z  (
 g `  w )  =  ( G `  (
 g  |`  w ) ) ) )   =>    |-  ( ph  ->  U. B : D --> S )
 
Theoremtfrcllembex 6261* Lemma for tfrcl 6267. The set  B exists. (Contributed by Jim Kingdon, 25-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `  g ) >. } ) ) }   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g : z --> S  /\  A. w  e.  z  (
 g `  w )  =  ( G `  (
 g  |`  w ) ) ) )   =>    |-  ( ph  ->  B  e.  _V )
 
Theoremtfrcllemubacc 6262* Lemma for tfrcl 6267. The union of  B satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `  g ) >. } ) ) }   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g : z --> S  /\  A. w  e.  z  (
 g `  w )  =  ( G `  (
 g  |`  w ) ) ) )   =>    |-  ( ph  ->  A. u  e.  D  ( U. B `  u )  =  ( G `  ( U. B  |`  u ) ) )
 
Theoremtfrcllemex 6263* Lemma for tfrcl 6267. (Contributed by Jim Kingdon, 26-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  B  =  { h  |  E. z  e.  D  E. g ( g : z --> S  /\  g  e.  A  /\  h  =  ( g  u.  { <. z ,  ( G `  g ) >. } ) ) }   &    |-  (
 ( ph  /\  x  e. 
 U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  A. z  e.  D  E. g ( g : z --> S  /\  A. w  e.  z  (
 g `  w )  =  ( G `  (
 g  |`  w ) ) ) )   =>    |-  ( ph  ->  E. f
 ( f : D --> S  /\  A. u  e.  D  ( f `  u )  =  ( G `  ( f  |`  u ) ) ) )
 
Theoremtfrcllemaccex 6264* We can define an acceptable function on any element of  X.

As with many of the transfinite recursion theorems, we have hypotheses that state that  F is a function and that it is defined up to  X. (Contributed by Jim Kingdon, 26-Mar-2022.)

 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   =>    |-  ( ( ph  /\  C  e.  X )  ->  E. g
 ( g : C --> S  /\  A. u  e.  C  ( g `  u )  =  ( G `  ( g  |`  u ) ) ) )
 
Theoremtfrcllemres 6265* Lemma for tfr1on 6253. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  A  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  Y 
 C_  dom  F )
 
Theoremtfrcldm 6266* Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  Y  e.  U. X )   =>    |-  ( ph  ->  Y  e.  dom  F )
 
Theoremtfrcl 6267* Closure for transfinite recursion. As with tfr1on 6253, the characteristic function must be defined up to a suitable point, not necessarily on all ordinals. (Contributed by Jim Kingdon, 25-Mar-2022.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  Ord 
 X )   &    |-  ( ( ph  /\  x  e.  X  /\  f : x --> S ) 
 ->  ( G `  f
 )  e.  S )   &    |-  ( ( ph  /\  x  e.  U. X )  ->  suc  x  e.  X )   &    |-  ( ph  ->  Y  e.  U. X )   =>    |-  ( ph  ->  ( F `  Y )  e.  S )
 
Theoremtfri1 6268* Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that  G is defined "everywhere", which is stated here as  ( G `  x )  e.  _V. Alternately,  A. x  e.  On A. f ( f  Fn  x  -> 
f  e.  dom  G
) would suffice.

Given a function  G satisfying that condition, we 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 Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

 |-  F  = recs ( G )   &    |-  ( Fun  G  /\  ( G `  x )  e.  _V )   =>    |-  F  Fn  On
 
Theoremtfri2 6269* Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule  G ( as described at tfri1 6268). Here we show that the function  F has the property that for any function  G satisfying that condition, the "next" value of  F is  G recursively applied to all "previous" values of  F. (Contributed by Jim Kingdon, 4-May-2019.)
 |-  F  = recs ( G )   &    |-  ( Fun  G  /\  ( G `  x )  e.  _V )   =>    |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
 
Theoremtfri3 6270* Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule  G ( as described at tfri1 6268). 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 Jim Kingdon, 4-May-2019.)
 |-  F  = recs ( G )   &    |-  ( Fun  G  /\  ( G `  x )  e.  _V )   =>    |-  (
 ( B  Fn  On  /\ 
 A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
 
Theoremtfrex 6271* The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  F  = recs ( G )   &    |-  ( ph  ->  A. x ( Fun  G  /\  ( G `  x )  e.  _V )
 )   =>    |-  ( ( ph  /\  A  e.  V )  ->  ( F `  A )  e. 
 _V )
 
2.6.20  Recursive definition generator
 
Syntaxcrdg 6272 Extend class notation with the recursive definition generator, with characteristic function  F and initial value  I.
 class  rec ( F ,  I
 )
 
Definitiondf-irdg 6273* Define a recursive definition generator on  On (the class of ordinal numbers) with characteristic function  F and initial value  I. 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 6208 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple. In classical logic it would be easier to divide this definition into cases based on whether the domain of  g is zero, a successor, or a limit ordinal. Cases do not (in general) work that way in intuitionistic logic, so instead we choose a definition which takes the union of all the results of the characteristic function for ordinals in the domain of  g. This means that this definition has the expected properties for increasing and continuous ordinal functions, which include ordinal addition and multiplication.

For finite recursion we also define df-frec 6294 and for suitable characteristic functions df-frec 6294 yields the same result as  rec restricted to  om, as seen at frecrdg 6311.

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 Jim Kingdon, 19-May-2019.)

 |- 
 rec ( F ,  I )  = recs (
 ( g  e.  _V  |->  ( I  u.  U_ x  e.  dom  g ( F `
  ( g `  x ) ) ) ) )
 
Theoremrdgeq1 6274 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 6275 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 ) )
 
Theoremrdgfun 6276 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |- 
 Fun  rec ( F ,  A )
 
Theoremrdgtfr 6277* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)
 |-  ( ( A. z
 ( F `  z
 )  e.  _V  /\  A  e.  V )  ->  ( Fun  ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `  ( g `  x ) ) ) ) 
 /\  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `  ( g `  x ) ) ) ) `
  f )  e. 
 _V ) )
 
Theoremrdgruledefgg 6278* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
 |-  ( ( F  Fn  _V 
 /\  A  e.  V )  ->  ( Fun  (
 g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
  ( g `  x ) ) ) )  /\  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
  ( g `  x ) ) ) ) `  f )  e.  _V ) )
 
Theoremrdgruledefg 6279* The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
 |-  F  Fn  _V   =>    |-  ( A  e.  V  ->  ( Fun  (
 g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
  ( g `  x ) ) ) )  /\  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
  ( g `  x ) ) ) ) `  f )  e.  _V ) )
 
Theoremrdgexggg 6280 The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
 |-  ( ( F  Fn  _V 
 /\  A  e.  V  /\  B  e.  W ) 
 ->  ( rec ( F ,  A ) `  B )  e.  _V )
 
Theoremrdgexgg 6281 The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
 |-  F  Fn  _V   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( rec ( F ,  A ) `  B )  e. 
 _V )
 
Theoremrdgifnon 6282 The recursive definition generator is a function on ordinal numbers. The  F  Fn  _V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6289; in cases like df-oadd 6323 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
 |-  ( ( F  Fn  _V 
 /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
 
Theoremrdgifnon2 6283* The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
 |-  ( ( A. z
 ( F `  z
 )  e.  _V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
 
Theoremrdgivallem 6284* Value of the recursive definition generator. Lemma for rdgival 6285 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
 |-  ( ( F  Fn  _V 
 /\  A  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  =  ( A  u.  U_ x  e.  B  ( F `  ( ( rec ( F ,  A )  |`  B ) `
  x ) ) ) )
 
Theoremrdgival 6285* Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)
 |-  ( ( F  Fn  _V 
 /\  A  e.  V  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  =  ( A  u.  U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x ) ) ) )
 
Theoremrdgss 6286 Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
 |-  ( ph  ->  F  Fn  _V )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  ( rec ( F ,  I ) `  A )  C_  ( rec ( F ,  I ) `  B ) )
 
Theoremrdgisuc1 6287* One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function  F other than  F  Fn  _V. Given that, the resulting expression encompasses both the expected successor term  ( F `  ( rec ( F ,  A ) `  B
) ) but also terms that correspond to the initial value  A and to limit ordinals  U_ x  e.  B ( F `  ( rec ( F ,  A ) `  x
) ).

If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6288. (Contributed by Jim Kingdon, 9-Jun-2019.)

 |-  ( ph  ->  F  Fn  _V )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  On )   =>    |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( A  u.  ( U_ x  e.  B  ( F `  ( rec ( F ,  A ) `  x ) )  u.  ( F `  ( rec ( F ,  A ) `  B ) ) ) ) )
 
Theoremrdgisucinc 6288* Value of the recursive definition generator at a successor.

This can be thought of as a generalization of oasuc 6366 and omsuc 6374. (Contributed by Jim Kingdon, 29-Aug-2019.)

 |-  ( ph  ->  F  Fn  _V )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  On )   &    |-  ( ph  ->  A. x  x  C_  ( F `  x ) )   =>    |-  ( ph  ->  ( rec ( F ,  A ) `  suc  B )  =  ( F `
  ( rec ( F ,  A ) `  B ) ) )
 
Theoremrdgon 6289* Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) (Revised by Jim Kingdon, 13-Apr-2022.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. x  e.  On  ( F `  x )  e. 
 On )   =>    |-  ( ( ph  /\  B  e.  On )  ->  ( rec ( F ,  A ) `  B )  e. 
 On )
 
Theoremrdg0 6290 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
 
Theoremrdg0g 6291 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
 |-  ( A  e.  C  ->  ( rec ( F ,  A ) `  (/) )  =  A )
 
Theoremrdgexg 6292 The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
 |-  A  e.  _V   &    |-  F  Fn  _V   =>    |-  ( B  e.  V  ->  ( rec ( F ,  A ) `  B )  e.  _V )
 
2.6.21  Finite recursion
 
Syntaxcfrec 6293 Extend class notation with the finite recursive definition generator, with characteristic function  F and initial value  I.
 class frec ( F ,  I )
 
Definitiondf-frec 6294* Define a recursive definition generator on  om (the class of finite ordinals) with characteristic function  F and initial value  I. 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 frec operation (especially when df-recs 6208 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple; see frec0g 6300 and frecsuc 6310.

Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4524. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 6311, this definition and df-irdg 6273 restricted to  om produce the same result.

Note: We introduce frec 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 Mario Carneiro and Jim Kingdon, 10-Aug-2019.)

 |- frec
 ( F ,  I
 )  =  (recs (
 ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `  m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) ) }
 ) )  |`  om )
 
Theoremfreceq1 6295 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
 |-  ( F  =  G  -> frec ( F ,  A )  = frec ( G ,  A ) )
 
Theoremfreceq2 6296 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
 |-  ( A  =  B  -> frec ( F ,  A )  = frec ( F ,  B ) )
 
Theoremfrecex 6297 Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
 |- frec
 ( F ,  A )  e.  _V
 
Theoremfrecfun 6298 Finite recursion produces a function. See also frecfnom 6304 which also states that the domain of that function is  om but which puts conditions on  A and  F. (Contributed by Jim Kingdon, 13-Feb-2022.)
 |- 
 Fun frec ( F ,  A )
 
Theoremnffrec 6299 Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ xfrec ( F ,  A )
 
Theoremfrec0g 6300 The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
 |-  ( A  e.  V  ->  (frec ( F ,  A ) `  (/) )  =  A )
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