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Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremriotasvd 6301* Deduction version of riotasv 6306. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  D  e.  A )   =>    |-  ( ( ph  /\  A  e.  V )  ->  (
 ( y  e.  B  /\  ps )  ->  D  =  C ) )
 
TheoremriotasvdOLD 6302* Deduction version of riotasv 6306. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  D  =  (
 iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   =>    |-  ( ( ( ph  /\  A  e.  V ) 
 /\  D  e.  A  /\  ( y  e.  B  /\  ps ) )  ->  D  =  C )
 
Theoremriotasv2d 6303* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4498). Special case of riota2f 6280. (Contributed by NM, 2-Mar-2013.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ y F )   &    |-  ( ph  ->  F/ y ch )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ( ph  /\  y  =  E ) 
 ->  ( ps  <->  ch ) )   &    |-  (
 ( ph  /\  y  =  E )  ->  C  =  F )   &    |-  ( ph  ->  D  e.  A )   &    |-  ( ph  ->  E  e.  B )   &    |-  ( ph  ->  ch )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  D  =  F )
 
Theoremriotasv2dOLD 6304* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4498). Special case of riota2f 6280. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( z  e.  F  ->  A. y  z  e.  F ) )   &    |-  ( ph  ->  ( ch  ->  A. y ch )
 )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  ( y  =  E  ->  ( ps  <->  ch ) ) )   &    |-  ( ph  ->  ( y  =  E  ->  C  =  F ) )   =>    |-  ( ( (
 ph  /\  A  e.  V )  /\  ( D  e.  A  /\  E  e.  B  /\  ch )
 )  ->  D  =  F )
 
Theoremriotasv2s 6305* The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4498) in the form of a substitution instance. Special case of riota2f 6280. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )   =>    |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  [_ E  /  y ]_ C )
 
Theoremriotasv 6306* Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4498). Special case of riota2f 6280. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  A  e.  _V   &    |-  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )   =>    |-  ( ( D  e.  A  /\  y  e.  B  /\  ph )  ->  D  =  C )
 
Theoremriotasv3d 6307* A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4498) also holds for its description binder  D (in the form of property  th). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ y th )   &    |-  ( ph  ->  D  =  (
 iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ( ph  /\  C  =  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )   &    |-  ( ph  ->  D  e.  A )   &    |-  ( ph  ->  E. y  e.  B  ps )   =>    |-  ( ( ph  /\  A  e.  V ) 
 ->  th )
 
Theoremriotasv3dOLD 6308* A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4498) also holds for its description binder  D (in the form of property  th). (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  A. y ph )   &    |-  ( ph  ->  ( th  ->  A. y th ) )   &    |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )   &    |-  ( ph  ->  ( C  =  D  ->  ( ch  <->  th ) ) )   &    |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )   =>    |-  ( ( ph  /\  ( A  e.  V  /\  D  e.  A  /\  E. y  e.  B  ps ) )  ->  th )
 
2.4.18  Functions on ordinals; strictly monotone ordinal functions
 
Theoremiunon 6309* 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 6310* 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   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  e.  On  -> 
 U_ x  e.  A  B  e.  On )
 
Theoremiinon 6311* 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 6312* 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 6313* A variant of onfununi 6312 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 6314* A variant of onovuni 6313 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 6315* There are no length  om decreasing sequences in the ordinals. See also noinfep 7314 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 6316 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
 wff  Smo  A
 
Definitiondf-smo 6317* 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 6318* 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 6319* 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 6320* 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 6321 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( A  =  B  ->  ( Smo  A  <->  Smo  B ) )
 
Theoremsmodm 6322 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( Smo  A  ->  Ord 
 dom  A )
 
Theoremsmores 6323 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 6324 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 6325 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 6326 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 6327 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 6328 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 6329 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 6330 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 6331 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 6332* 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 6333 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 6334 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 6335 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 6336 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 6337 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 6338 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 6339 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 6340 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 ) ) )
 
2.4.19  "Strong" transfinite recursion
 
Syntaxcrecs 6341 Notation for a function defined by strong transfinite recursion.
 class recs ( F )
 
Definitiondf-recs 6342* 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 6377 for more details on why this definition is desirable. Unlike df-rdg 6377 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 6370 and recsval 6371 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7201, zorn2 8087, and dfac8alem 7610. (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 6343 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
 
Theoremnfrecs 6344 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F/_ x F   =>    |-  F/_ xrecs ( F )
 
Theoremtfrlem1 6345* 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 6346* Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6345 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 6347* 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 6348* 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 6349* 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 6350* 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 6351* 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 6352* 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 6353* 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 6354* 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 6355* 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 6356* Lemma for transfinite recursion. Without using ax-rep 4091, 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 6357* 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 6361, 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 6358* 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 6359* 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 6360* 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 6361* Lemma for transfinite recursion. Assuming ax-rep 4091,  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 6362* Lemma for transfinite recursion. Without assuming ax-rep 4091, 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 6363* Lemma for finite recursion. Without assuming ax-rep 4091, 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 6364 A weak version of tfr1 6367 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 6365 A weak version of tfr2 6368 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 6366 Without assuming ax-rep 4091, 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 6367 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 6368 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 6369* 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 6370 Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |- recs
 ( F )  Fn 
 On
 
Theoremrecsval 6371 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 6372*  G is a function. Lemma for tz7.44-1 6373, tz7.44-2 6374, and tz7.44-3 6375. (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 6373* 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 6374* 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 6375* 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 ) )
 
2.4.20  Recursive definition generator
 
Syntaxcrdg 6376 Extend class notation with the recursive definition generator, with characteristic function  F and initial value  I.
 class  rec ( F ,  I
 )
 
Definitiondf-rdg 6377* 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 6367 and  G in tz7.44-1 6373 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 6464, from which we prove the recursive textbook definition as theorems oa0 6469, oasuc 6477, and oalim 6485 (with the help of theorems rdg0 6388, rdgsuc 6391, and rdglim2a 6400). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6402 and frsuc 6403. 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 3526) 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 10999 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 11241 and integer powers df-exp 11057.

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 6378 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 6379 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 6380 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 6381 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 6382* 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 6383 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |- 
 Fun  rec ( F ,  A )
 
Theoremrdgdmlim 6384 The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)
 |- 
 Lim  dom  rec ( F ,  A )
 
Theoremrdgfnon 6385 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 6386* 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 6387* 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 6388 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 6389 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 6390 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 6391 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 6392 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 6393 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 6394 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
 |-  ( A  e.  C  ->  ( rec ( F ,  A ) `  (/) )  =  A )
 
Theoremrdgsucmptf 6395 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 6396 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 6395 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 6397* This version of rdgsucmpt 6398 avoids the not-free hypothesis of rdgsucmptf 6395 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 6398* 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 6399* 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 6400* 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 ) )
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