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Theorem List for Metamath Proof Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtfrlem12 6401* 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 6402* 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 6403* Lemma for transfinite recursion. Assuming ax-rep 4133,  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 6404* Lemma for transfinite recursion. Without assuming ax-rep 4133, 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 6405* Lemma for finite recursion. Without assuming ax-rep 4133, 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 6406 A weak version of tfr1 6409 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 6407 A weak version of tfr2 6410 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 6408 Without assuming ax-rep 4133, 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 6409 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 6410 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 6411* 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 6412 Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |- recs
 ( F )  Fn 
 On
 
Theoremrecsval 6413 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 6414*  G is a function. Lemma for tz7.44-1 6415, tz7.44-2 6416, and tz7.44-3 6417. (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 6415* 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 6416* 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 6417* 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 ) )
 
2.4.21  Recursive definition generator
 
Syntaxcrdg 6418 Extend class notation with the recursive definition generator, with characteristic function  F and initial value  I.
 class  rec ( F ,  I
 )
 
Definitiondf-rdg 6419* 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 6409 and  G in tz7.44-1 6415 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 6506, from which we prove the recursive textbook definition as theorems oa0 6511, oasuc 6519, and oalim 6527 (with the help of theorems rdg0 6430, rdgsuc 6433, and rdglim2a 6442). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6444 and frsuc 6445. 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 3568) 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 11042 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 11284 and integer powers df-exp 11100.

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 6420 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 6421 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 6422 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 6423 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 6424* 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 6425 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |- 
 Fun  rec ( F ,  A )
 
Theoremrdgdmlim 6426 The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)
 |- 
 Lim  dom  rec ( F ,  A )
 
Theoremrdgfnon 6427 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 6428* 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 6429* 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 6430 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 6431 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 6432 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 6433 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 6434 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 6435 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 6436 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
 |-  ( A  e.  C  ->  ( rec ( F ,  A ) `  (/) )  =  A )
 
Theoremrdgsucmptf 6437 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 6438 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 6437 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 6439* This version of rdgsucmpt 6440 avoids the not-free hypothesis of rdgsucmptf 6437 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 6440* 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 6441* 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 6442* 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 ) )
 
2.4.22  Finite recursion
 
Theoremfrfnom 6443 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 6444 The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.)
 |-  ( A  e.  B  ->  ( ( rec ( F ,  A )  |` 
 om ) `  (/) )  =  A )
 
Theoremfrsuc 6445 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 6446 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 6447 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 6446 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 6448* 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 6449* 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 6450* 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 6451* 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 )
 
Theoremtz7.48-3 6452* Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
 |-  F  Fn  On   =>    |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  A  e.  _V )
 
Theoremtz7.49 6453* Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 10-Jan-2013.)
 |-  F  Fn  On   &    |-  ( ph 
 <-> 
 A. x  e.  On  ( ( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )   =>    |-  ( ( A  e.  B  /\  ph )  ->  E. x  e.  On  ( A. y  e.  x  ( A  \  ( F " y
 ) )  =/=  (/)  /\  ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) ) )
 
Theoremtz7.49c 6454* Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
 |-  F  Fn  On   =>    |-  ( ( A  e.  B  /\  A. x  e.  On  (
 ( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
 
Syntaxcseqom 6455 Extend class notation to include index-aware recursive definitions.
 class seq𝜔 ( F ,  I )
 
Definitiondf-seqom 6456* Index-aware recursive definitions over  om. A mashup of df-rdg 6419 and df-seq 11042, this allows for recursive definitions that use an index in the recursion in cases where Infinity is not admitted. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |- seq𝜔 ( F ,  I )  =  ( rec ( ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I
 ) >. ) " om )
 
Theoremseqomlem0 6457* Lemma for seq𝜔. Change bound variables. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |- 
 rec ( ( a  e.  om ,  b  e.  _V  |->  <. suc  a ,  ( a F b ) >. ) ,  <. (/) ,  (  _I  `  I
 ) >. )  =  rec ( ( c  e. 
 om ,  d  e. 
 _V  |->  <. suc  c ,  ( c F d ) >. ) ,  <. (/) ,  (  _I  `  I
 ) >. )
 
Theoremseqomlem1 6458* Lemma for seq𝜔. The underlying recursion generates a sequence of pairs with the expected first values. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
 |-  Q  =  rec (
 ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I ) >. )   =>    |-  ( A  e.  om 
 ->  ( Q `  A )  =  <. A ,  ( 2nd `  ( Q `  A ) ) >. )
 
Theoremseqomlem2 6459* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
 |-  Q  =  rec (
 ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I ) >. )   =>    |-  ( Q " om )  Fn  om
 
Theoremseqomlem3 6460* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  Q  =  rec (
 ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I ) >. )   =>    |-  ( ( Q
 " om ) `  (/) )  =  (  _I  `  I )
 
Theoremseqomlem4 6461* Lemma for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
 |-  Q  =  rec (
 ( i  e.  om ,  v  e.  _V  |->  <. suc  i ,  ( i F v ) >. ) ,  <. (/) ,  (  _I  `  I ) >. )   =>    |-  ( A  e.  om 
 ->  ( ( Q " om ) `  suc  A )  =  ( A F ( ( Q
 " om ) `  A ) ) )
 
Theoremseqomeq12 6462 Equality theorem for seq𝜔. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  ( ( A  =  B  /\  C  =  D )  -> seq𝜔 ( A ,  C )  = seq𝜔 ( B ,  D ) )
 
Theoremfnseqom 6463 An index-aware recursive definition defines a function on the natural numbers. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  G  = seq𝜔 ( F ,  I
 )   =>    |-  G  Fn  om
 
Theoremseqom0g 6464 Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  G  = seq𝜔 ( F ,  I
 )   =>    |-  ( I  e.  _V  ->  ( G `  (/) )  =  I )
 
Theoremseqomsuc 6465 Value of an index-aware recursive definition at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)
 |-  G  = seq𝜔 ( F ,  I
 )   =>    |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( A F ( G `  A ) ) )
 
2.4.23  Abian's "most fundamental" fixed point theorem
 
Theoremabianfplem 6466* Lemma for abianfp 6467. We prove by transfinite induction that if  F has a fixed point  x, then its iterates also equal  x. This lemma is used for the "trivial" direction of the main theorem. (Contributed by NM, 4-Sep-2004.)
 |-  A  e.  _V   &    |-  G  =  rec ( ( z  e.  _V  |->  ( F `
  z ) ) ,  x )   =>    |-  ( v  e. 
 On  ->  ( ( F `
  x )  =  x  ->  ( G `  v )  =  x ) )
 
Theoremabianfp 6467* "A most fundamental fixed point theorem" of Alexander Abian (1923-1999), apparently proved in 1998. Let  G `  0  =  x,  G `  1  =  F `  x,  G `  2  =  F `  ( F `
 x ),... be the iterates of  F. The theorem reads (using our variable names): "Let  F be a mapping from a set  A into itself. Then  F has a fixed point if and only if: There exists an element  x of  A such that for every ordinal  v,  G `  v is an element of  A, and if  G `  v is not a fixed point of  F then the  G `  u's are all distinct for every ordinal  u  e.  v." See df-rdg 6419 for the  rec operation. The proof's key idea is to assume that  F does not have a fixed point, then use the Axiom of Replacement in the form of f1dmex 5713 to derive that the class of all ordinal numbers exists, contradicting onprc 4576. Our version of this theorem does not require the hypothesis that  F be a mapping. Reference: http://us2.metamath.org:88/abian-themostfixed.html. For an application of this theorem, see http://groups.google.com/group/sci.stat.math/msg/1737ee1133c24aeb for its use in a proof of Tarski's fixed point theorem. (Contributed by NM, 5-Sep-2004.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  A  e.  _V   &    |-  G  =  rec ( ( z  e.  _V  |->  ( F `
  z ) ) ,  x )   =>    |-  ( E. x  e.  A  ( F `  x )  =  x  <->  E. x  e.  A  A. v  e.  On  (
 ( G `  v
 )  e.  A  /\  ( -.  ( F `  ( G `  v ) )  =  ( G `
  v )  ->  A. u  e.  v  -.  ( G `  v
 )  =  ( G `
  u ) ) ) )
 
2.4.24  Ordinal arithmetic
 
Syntaxc1o 6468 Extend the definition of a class to include the ordinal number 1.
 class  1o
 
Syntaxc2o 6469 Extend the definition of a class to include the ordinal number 2.
 class  2o
 
Syntaxc3o 6470 Extend the definition of a class to include the ordinal number 3.
 class  3o
 
Syntaxc4o 6471 Extend the definition of a class to include the ordinal number 4.
 class  4o
 
Syntaxcoa 6472 Extend the definition of a class to include the ordinal addition operation.
 class  +o
 
Syntaxcomu 6473 Extend the definition of a class to include the ordinal multiplication operation.
 class  .o
 
Syntaxcoe 6474 Extend the definition of a class to include the ordinal exponentiation operation.
 class  ^o
 
Definitiondf-1o 6475 Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  =  suc  (/)
 
Definitiondf-2o 6476 Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.)
 |- 
 2o  =  suc  1o
 
Definitiondf-3o 6477 Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |- 
 3o  =  suc  2o
 
Definitiondf-4o 6478 Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |- 
 4o  =  suc  3o
 
Definitiondf-oadd 6479* Define the ordinal addition operation. (Contributed by NM, 3-May-1995.)
 |- 
 +o  =  ( x  e.  On ,  y  e.  On  |->  ( rec (
 ( z  e.  _V  |->  suc  z ) ,  x ) `  y ) )
 
Definitiondf-omul 6480* Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.)
 |- 
 .o  =  ( x  e.  On ,  y  e.  On  |->  ( rec (
 ( z  e.  _V  |->  ( z  +o  x ) ) ,  (/) ) `  y ) )
 
Definitiondf-oexp 6481* Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)
 |- 
 ^o  =  ( x  e.  On ,  y  e.  On  |->  if ( x  =  (/) ,  ( 1o  \  y ) ,  ( rec ( ( z  e. 
 _V  |->  ( z  .o  x ) ) ,  1o ) `  y
 ) ) )
 
Theorem1on 6482 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  On
 
Theorem2on 6483 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |- 
 2o  e.  On
 
Theorem2on0 6484 Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.)
 |- 
 2o  =/=  (/)
 
Theorem3on 6485 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 3o  e.  On
 
Theorem4on 6486 Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |- 
 4o  e.  On
 
Theoremdf1o2 6487 Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.)
 |- 
 1o  =  { (/) }
 
Theoremdf2o3 6488 Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 2o  =  { (/) ,  1o }
 
Theoremdf2o2 6489 Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
 |- 
 2o  =  { (/) ,  { (/)
 } }
 
Theorem1n0 6490 Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.)
 |- 
 1o  =/=  (/)
 
Theoremxp01disj 6491 Cross products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
 |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
 ) )  =  (/)
 
Theoremordgt0ge1 6492 Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
 |-  ( Ord  A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
 
Theoremordge1n0 6493 An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.)
 |-  ( Ord  A  ->  ( 1o  C_  A  <->  A  =/=  (/) ) )
 
Theoremel1o 6494 Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  ( A  e.  1o  <->  A  =  (/) )
 
Theoremdif1o 6495 Two ways to say that  A is a nonzero number of the set  B. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  ( A  e.  ( B  \  1o )  <->  ( A  e.  B  /\  A  =/=  (/) ) )
 
Theoremondif1 6496 Two ways to say that  A is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  ( A  e.  ( On  \  1o )  <->  ( A  e.  On  /\  (/)  e.  A ) )
 
Theoremondif2 6497 Two ways to say that  A is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)
 |-  ( A  e.  ( On  \  2o )  <->  ( A  e.  On  /\  1o  e.  A ) )
 
Theorem2oconcl 6498 Closure of the pair swapping function on  2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( A  e.  2o  ->  ( 1o  \  A )  e.  2o )
 
Theorem0lt1o 6499 Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.)
 |-  (/)  e.  1o
 
Theoremdif20el 6500 An ordinal greater than one is greater than zero. (Contributed by Mario Carneiro, 25-May-2015.)
 |-  ( A  e.  ( On  \  2o )  ->  (/) 
 e.  A )
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