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Theorem List for Metamath Proof Explorer - 25501-25600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtrpredeq2 25501 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  =  B  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )
 
Theoremtrpredeq3 25502 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( X  =  Y  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  A ,  Y ) )
 
Theoremtrpredeq1d 25503 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( ph  ->  R  =  S )   =>    |-  ( ph  ->  TrPred ( R ,  A ,  X )  =  TrPred ( S ,  A ,  X ) )
 
Theoremtrpredeq2d 25504 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )
 
Theoremtrpredeq3d 25505 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( ph  ->  X  =  Y )   =>    |-  ( ph  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  A ,  Y ) )
 
Theoremeltrpred 25506* A class is a transitive predecessor iff it is in some value of the underlying function. This theorem is not really meant to be used directly: instead refer to trpredpred 25508 and trpredmintr 25511. (Contributed by Scott Fenton, 28-Apr-2012.)
 |-  ( Y  e.  TrPred ( R ,  A ,  X ) 
 <-> 
 E. i  e.  om  Y  e.  ( ( rec ( ( a  e. 
 _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) , 
 Pred ( R ,  A ,  X )
 )  |`  om ) `  i ) )
 
Theoremtrpredlem1 25507* Technical lemma for transitive predecessors properties. All values of the transitive predecessors' underlying function are subsets of the base set. (Contributed by Scott Fenton, 28-Apr-2012.)
 |-  ( Pred ( R ,  A ,  X )  e.  B  ->  ( ( rec (
 ( a  e.  _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) , 
 Pred ( R ,  A ,  X )
 )  |`  om ) `  i )  C_  A )
 
Theoremtrpredpred 25508 Assuming it exists, the predecessor class is a subset of the transitive predecessors. (Contributed by Scott Fenton, 18-Feb-2011.)
 |-  ( Pred ( R ,  A ,  X )  e.  B  -> 
 Pred ( R ,  A ,  X )  C_  TrPred ( R ,  A ,  X ) )
 
Theoremtrpredss 25509 The transitive predecessors form a subset of the base class. (Contributed by Scott Fenton, 20-Feb-2011.)
 |-  ( Pred ( R ,  A ,  X )  e.  B  -> 
 TrPred ( R ,  A ,  X )  C_  A )
 
Theoremtrpredtr 25510 The transitive predecessors are transitive in  R and 
A (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  ( Y  e.  TrPred ( R ,  A ,  X )  ->  Pred ( R ,  A ,  Y )  C_  TrPred ( R ,  A ,  X ) ) )
 
Theoremtrpredmintr 25511* The transitive predecessors form the smallest class transitive in  R and  A. That is, if  B is another  R,  A transitive class containing  Pred ( R ,  A ,  X ), then  TrPred ( R ,  A ,  X )  C_  B (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( X  e.  A  /\  R Se  A ) 
 /\  ( A. y  e.  B  Pred ( R ,  A ,  y )  C_  B  /\  Pred ( R ,  A ,  X )  C_  B ) )  ->  TrPred ( R ,  A ,  X )  C_  B )
 
Theoremtrpredelss 25512 Given a transitive predecessor  Y of  X, the transitive predecessors of  Y are a subset of the transitive predecessors of  X. (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  ( Y  e.  TrPred ( R ,  A ,  X )  ->  TrPred ( R ,  A ,  Y )  C_  TrPred ( R ,  A ,  X ) ) )
 
Theoremdftrpred3g 25513* The transitive predecessors of  X are equal to the predecessors of  X together with their transitive predecessors. (Contributed by Scott Fenton, 26-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  TrPred ( R ,  A ,  X )  =  ( Pred ( R ,  A ,  X )  u.  U_ y  e.  Pred  ( R ,  A ,  X ) TrPred ( R ,  A ,  y )
 ) )
 
Theoremdftrpred4g 25514* Another recursive expression for the transitive predecessors. (Contributed by Scott Fenton, 27-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  TrPred ( R ,  A ,  X )  =  U_ y  e.  Pred  ( R ,  A ,  X )
 ( { y }  u.  TrPred ( R ,  A ,  y )
 ) )
 
Theoremtrpredpo 25515 If  R partially orders  A, then the transitive predecessors are the same as the immediate predecessors . (Contributed by Scott Fenton, 28-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( R  Po  A  /\  X  e.  A  /\  R Se  A )  ->  TrPred ( R ,  A ,  X )  =  Pred ( R ,  A ,  X ) )
 
Theoremtrpred0 25516 The class of transitive predecessors is empty when  A is empty. (Contributed by Scott Fenton, 30-Apr-2012.)
 |-  TrPred ( R ,  (/) ,  X )  =  (/)
 
Theoremtrpredex 25517 The transitive predecessors of a relation form a set (NOTE: this is the first theorem in the transitive predecessor series that requires infinity). (Contributed by Scott Fenton, 18-Feb-2011.)
 |-  TrPred ( R ,  A ,  X )  e.  _V
 
Theoremtrpredrec 25518* If  Y is an  R,  A transitive predecessor, then it is either an immediate predecessor or there is a transitive predecessor between  Y and  X (Contributed by Scott Fenton, 9-May-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( X  e.  A  /\  R Se  A )  ->  ( Y  e.  TrPred ( R ,  A ,  X )  ->  ( Y  e.  Pred
 ( R ,  A ,  X )  \/  E. z  e.  TrPred  ( R ,  A ,  X ) Y R z ) ) )
 
19.7.24  Founded Induction
 
Theoremfrmin 25519* Every (possibly proper) subclass of a class  A with a founded, set-like relation  R has a minimal element. Lemma 4.3 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory. This is a very strong generalization of tz6.26 25482 and tz7.5 4604. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  Fr  A  /\  R Se  A ) 
 /\  ( B  C_  A  /\  B  =/=  (/) ) ) 
 ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
 
Theoremfrind 25520* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 25519). This principle states that if  B is a subclass of a founded class  A with the property that every element of  B whose initial segment is included in  A is itself equal to  A. Compare wfi 25484 and tfi 4835, which are special cases of this theorem that do not require the axiom of infinity to prove. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  Fr  A  /\  R Se  A ) 
 /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  A  =  B )
 
Theoremfrindi 25521* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 25519). This principle states that if  B is a subclass of a founded class  A with the property that every element of  B whose initial segment is included in  A is itself equal to  A. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   =>    |-  ( ( B 
 C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  =  B )
 
Theoremfrinsg 25522* Founded Induction Schema. If a property passes from all elements less than  y of a founded class  A to  y itself (induction hypothesis), then the property holds for all elements of  A. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) [. z  /  y ]. ph  ->  ph ) )   =>    |-  ( ( R  Fr  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremfrins 25523* Founded Induction Schema. If a property passes from all elements less than  y of a founded class  A to  y itself (induction hypothesis), then the property holds for all elements of  A. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) [. z  /  y ]. ph  ->  ph ) )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremfrins2fg 25524* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   &    |-  F/ y ps   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   =>    |-  ( ( R  Fr  A  /\  R Se  A )  ->  A. y  e.  A  ph )
 
Theoremfrins2f 25525* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  F/ y ps   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   &    |-  ( y  e.  A  ->  ( A. z  e.  Pred  ( R ,  A ,  y
 ) ps  ->  ph )
 )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremfrins2g 25526* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 8-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   =>    |-  ( ( R  Fr  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremfrins2 25527* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  (
 y  =  z  ->  ( ph  <->  ps ) )   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremfrins3 25528* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  (
 y  =  z  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  (
 ph 
 <->  ch ) )   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   =>    |-  ( B  e.  A  ->  ch )
 
19.7.25  Ordering Ordinal Sequences
 
Theoremorderseqlem 25529* Lemma for poseq 25530 and soseq 25531. The function value of a sequene is either in  A or null. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  F  =  { f  |  E. x  e.  On  f : x --> A }   =>    |-  ( G  e.  F  ->  ( G `  X )  e.  ( A  u.  { (/) } )
 )
 
Theoremposeq 25530* A partial ordering of sequences of ordinals. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  R  Po  ( A  u.  { (/)
 } )   &    |-  F  =  {
 f  |  E. x  e.  On  f : x --> A }   &    |-  S  =  { <. f ,  g >.  |  ( ( f  e.  F  /\  g  e.  F )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
 )  =  ( g `
  y )  /\  ( f `  x ) R ( g `  x ) ) ) }   =>    |-  S  Po  F
 
Theoremsoseq 25531* A linear ordering of sequences of ordinals. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  R  Or  ( A  u.  { (/)
 } )   &    |-  F  =  {
 f  |  E. x  e.  On  f : x --> A }   &    |-  S  =  { <. f ,  g >.  |  ( ( f  e.  F  /\  g  e.  F )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
 )  =  ( g `
  y )  /\  ( f `  x ) R ( g `  x ) ) ) }   &    |-  -.  (/)  e.  A   =>    |-  S  Or  F
 
19.7.26  Well-founded recursion
 
Syntaxcwrecs 25532 Declare syntax for the well-founded recusive function generator.
 class wrecs ( R ,  A ,  F )
 
Definitiondf-wrecs 25533* Here we define the well-founded recusive function generator. This is similar to recs, but works with arbitrary well-founded relationships. (Contributed by Scott Fenton, 7-Jun-2018.) (New usage is discouraged.)
 |- wrecs ( R ,  A ,  F )  =  U. { f  |  E. x ( f  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }
 
Theoremwrecseq123 25534 General equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
 |-  (
 ( R  =  S  /\  A  =  B  /\  F  =  G )  -> wrecs ( R ,  A ,  F )  = wrecs ( S ,  B ,  G ) )
 
Theoremnfwrecs 25535 Bound-variable hypothesis builder for the well-founded recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.)
 |-  F/_ x R   &    |-  F/_ x A   &    |-  F/_ x F   =>    |-  F/_ xwrecs ( R ,  A ,  F )
 
Theoremwrecseq1 25536 Equality theorem for the well-founded recusive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
 |-  ( R  =  S  -> wrecs ( R ,  A ,  F )  = wrecs ( S ,  A ,  F ) )
 
Theoremwrecseq2 25537 Equality theorem for the well-founded recusive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
 |-  ( A  =  B  -> wrecs ( R ,  A ,  F )  = wrecs ( R ,  B ,  F ) )
 
Theoremwrecseq3 25538 Equality theorem for the well-founded recusive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
 |-  ( F  =  G  -> wrecs ( R ,  A ,  F )  = wrecs ( R ,  A ,  G ) )
 
Theoremwfr3g 25539* Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  (
 ( ( R  We  A  /\  R Se  A ) 
 /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A ,  y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
 ) ) ) ) 
 ->  F  =  G )
 
Theoremwfrlem1 25540* Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions  B. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  B  =  { g  |  E. z ( g  Fn  z  /\  (
 z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z ) 
 /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  Pred ( R ,  A ,  w ) ) ) ) }
 
Theoremwfrlem2 25541* Lemma for well-founded recursion. An acceptable function is a function. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( g  e.  B  ->  Fun  g )
 
Theoremwfrlem3 25542* Lemma for well-founded recursion. An acceptable function's domain is a subset of  A. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( g  e.  B  ->  dom  g  C_  A )
 
Theoremwfrlem4 25543* Lemma for well-founded recursion. Properties of the restriction of an acceptable function to the domain of another one. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  B  =  {
 f  |  E. x ( f  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x )  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( g  |`  ( dom  g  i^i  dom  h ) )  Fn  ( dom  g  i^i  dom  h )  /\  A. a  e.  ( dom  g  i^i 
 dom  h ) ( ( g  |`  ( dom  g  i^i  dom  h ) ) `  a
 )  =  ( F `
  ( ( g  |`  ( dom  g  i^i 
 dom  h ) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
 
Theoremwfrlem5 25544* Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( x g u 
 /\  x h v )  ->  u  =  v ) )
 
Theoremwfrlem6 25545 Lemma for well-founded recursion. The definition generates a relationship. (Contributed by Scott Fenton, 8-Jun-2018.)
 |-  F  = wrecs ( R ,  A ,  G )   =>    |- 
 Rel  F
 
Theoremwfrlem7 25546 Lemma for well-founded recursion. The domain of  F is a subclass of  A. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  F  = wrecs ( R ,  A ,  G )   =>    |- 
 dom  F  C_  A
 
Theoremwfrlem8 25547 Lemma for well-founded recursion. Compute the prececessor class for an  R minimal element of  ( A  \  dom  F ). (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( Pred ( R ,  ( A  \  dom  F ) ,  X )  =  (/)  <->  Pred ( R ,  A ,  X )  =  Pred ( R ,  dom  F ,  X ) )
 
Theoremwfrlem9 25548 Lemma for well-founded recursion. If  X  e.  dom  F, then its predecessor class is a subset of  dom  F. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( X  e.  dom  F 
 ->  Pred ( R ,  A ,  X )  C_ 
 dom  F )
 
Theoremwfrlem10 25549* Lemma for well-founded recursion. When  z is an  R minimal element of  ( A  \  dom  F ), then its predecessor class is equal to  dom  F. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( ( z  e.  ( A  \  dom  F )  /\  Pred ( R ,  ( A  \ 
 dom  F ) ,  z
 )  =  (/) )  ->  Pred ( R ,  A ,  z )  =  dom  F )
 
Theoremwfrlem11 25550 Lemma for well-founded recursion. The union of all acceptable functions is a function. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |- 
 Fun  F
 
Theoremwfrlem12 25551* Lemma for well-founded recursion. Here, we compute the value of the recursive definition generator. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( y  e.  dom  F 
 ->  ( F `  y
 )  =  ( G `
  ( F  |`  Pred ( R ,  A ,  y ) ) ) )
 
Theoremwfrlem13 25552* Lemma for well-founded recursion. From here through wfrlem16 25555, we aim to prove that  dom  F  =  A. We do this by supposing that there is an element  z of  A that is not in  dom  F. We then define  C by extending  dom  F with the appropriate value at  z. We then show that  z cannot be an  R minimal element of  ( A  \  dom  F ), meaning that  ( A  \  dom  F ) must be empty, so  dom  F  =  A. Here, we show that  C is a function extending the domain of  F by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z ) ) )
 >. } )   =>    |-  ( z  e.  ( A  \  dom  F ) 
 ->  C  Fn  ( dom 
 F  u.  { z } ) )
 
Theoremwfrlem14 25553* Lemma for well-founded recursion. Compute the value of  C. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z ) ) )
 >. } )   =>    |-  ( z  e.  ( A  \  dom  F ) 
 ->  ( y  e.  ( dom  F  u.  { z } )  ->  ( C `
  y )  =  ( G `  ( C  |`  Pred ( R ,  A ,  y )
 ) ) ) )
 
Theoremwfrlem15 25554* Lemma for well-founded recursion. When  z is  R minimal,  C is an acceptable function. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z ) ) )
 >. } )   =>    |-  ( ( z  e.  ( A  \  dom  F )  /\  Pred ( R ,  ( A  \ 
 dom  F ) ,  z
 )  =  (/) )  ->  C  e.  { f  |  E. x ( f  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) } )
 
Theoremwfrlem16 25555* Lemma for well-founded recursion. If 
z is  R minimal in  ( A  \  dom  F ), then  C is acceptable and thus a subset of  F, but  dom  C is bigger than  dom  F. Thus, 
z cannot be minimal, so  ( A  \  dom  F ) must be empty, and (due to wfrlem7 25546),  dom  F  =  A. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z ) ) )
 >. } )   =>    |- 
 dom  F  =  A
 
Theoremwfr1 25556 The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function  G and a class of "acceptable" functions  B. Then, using a base class  A and a well-ordering  R of  A, we define a function  F. This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of  F. We begin by showing that  F is a function over  A. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  F  Fn  A
 
Theoremwfr2 25557 The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of  F at any  z  e.  A is  G recursively applied to all "previous" values of  F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( X  e.  A  ->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X ) ) ) )
 
Theoremwfr3 25558* The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that  F is unique. We do this by showing that any function  H with the same properties we proved of  F in wfr1 25556 and wfr2 25557 is identical to  F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( ( H  Fn  A  /\  A. z  e.  A  ( H `  z )  =  ( G `  ( H  |`  Pred ( R ,  A ,  z ) ) ) )  ->  F  =  H )
 
19.7.27  Transfinite recursion via Well-founded recursion
 
TheoremtfrALTlem 25559 Lemma for deriving transfinite recursion from well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.)
 |- recs ( G )  = wrecs (  _E 
 ,  On ,  G )
 
Theoremtfr1ALT 25560 tfr1 6660 via well-founded recursion. (Contributed by Scott Fenton, 17-Aug-1994.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  F  = recs ( G )   =>    |-  F  Fn  On
 
Theoremtfr2ALT 25561 tfr2 6661 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  F  = recs ( G )   =>    |-  ( z  e. 
 On  ->  ( F `  z )  =  ( G `  ( F  |`  z ) ) )
 
Theoremtfr3ALT 25562* tfr3 6662 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  F  = recs ( G )   =>    |-  ( ( B  Fn  On  /\  A. x  e.  On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
 
19.7.28  Well-founded zero, successor, and limits
 
Syntaxcwsuc 25563 Declare the syntax for well-founded successor.
 class wsuc ( R ,  A ,  X )
 
Syntaxcwlim 25564 Declare the syntax for well-founded limit class.
 class WLim ( R ,  A )
 
Definitiondf-wsuc 25565 Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.)
 |- wsuc ( R ,  A ,  X )  =  sup ( Pred ( `' R ,  A ,  X ) ,  A ,  `' R )
 
Definitiondf-wlim 25566* Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.)
 |- WLim ( R ,  A )  =  { x  e.  A  |  ( x  =/=  sup ( A ,  A ,  `' R )  /\  x  =  sup ( Pred ( R ,  A ,  x ) ,  A ,  R ) ) }
 
Theoremwsuceq123 25567 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  (
 ( R  =  S  /\  A  =  B  /\  X  =  Y )  -> wsuc ( R ,  A ,  X )  = wsuc ( S ,  B ,  Y ) )
 
Theoremwsuceq1 25568 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( R  =  S  -> wsuc ( R ,  A ,  X )  = wsuc ( S ,  A ,  X ) )
 
Theoremwsuceq2 25569 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( A  =  B  -> wsuc ( R ,  A ,  X )  = wsuc ( R ,  B ,  X ) )
 
Theoremwsuceq3 25570 Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( X  =  Y  -> wsuc ( R ,  A ,  X )  = wsuc ( R ,  A ,  Y ) )
 
Theoremnfwsuc 25571 Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  F/_ x R   &    |-  F/_ x A   &    |-  F/_ x X   =>    |-  F/_ xwsuc ( R ,  A ,  X )
 
Theoremwlimeq12 25572 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
 |-  (
 ( R  =  S  /\  A  =  B ) 
 -> WLim ( R ,  A )  = WLim ( S ,  B ) )
 
Theoremwlimeq1 25573 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
 |-  ( R  =  S  -> WLim ( R ,  A )  = WLim ( S ,  A ) )
 
Theoremwlimeq2 25574 Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
 |-  ( A  =  B  -> WLim ( R ,  A )  = WLim ( R ,  B ) )
 
Theoremnfwlim 25575 Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/_ xWLim ( R ,  A )
 
Theoremelwlim 25576 Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
 |-  ( X  e. WLim ( R ,  A )  <->  ( X  e.  A  /\  X  =/=  sup ( A ,  A ,  `' R )  /\  X  =  sup ( Pred ( R ,  A ,  X ) ,  A ,  R ) ) )
 
Theoremwzel 25577 The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  (
 ( R  We  A  /\  R Se  A  /\  A  =/= 
 (/) )  ->  sup ( A ,  A ,  `' R )  e.  A )
 
Theoremwsuclem 25578* Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.)
 |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  E. w  e.  A  X R w )   =>    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  Pred  ( `' R ,  A ,  X )  -.  x `' R y  /\  A. y  e.  A  (
 y `' R x 
 ->  E. z  e.  Pred  ( `' R ,  A ,  X ) y `' R z ) ) )
 
Theoremwsucex 25579 Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.)
 |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  -> wsuc ( R ,  A ,  X )  e.  _V )
 
Theoremwsuccl 25580* If  X is a set with an  R successor in  A, then its well-founded successor is a member of  A. (Contributed by Scott Fenton, 15-Jun-2018.)
 |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  E. y  e.  A  X R y )   =>    |-  ( ph  -> wsuc ( R ,  A ,  X )  e.  A )
 
Theoremwsuclb 25581 A well-founded successor is a lower bound on points after  X. (Contributed by Scott Fenton, 16-Jun-2018.)
 |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  A )   &    |-  ( ph  ->  X R Y )   =>    |-  ( ph  ->  -.  Y Rwsuc ( R ,  A ,  X ) )
 
Theoremwlimss 25582 The class of limit points is a subclass of the base class. (Contributed by Scott Fenton, 16-Jun-2018.)
 |- WLim ( R ,  A )  C_  A
 
19.7.29  Founded Recursion
 
Theoremfrr3g 25583* Functions defined by founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  Fr  A  /\  R Se  A ) 
 /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  (
 y H ( F  |`  Pred ( R ,  A ,  y )
 ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( y H ( G  |`  Pred ( R ,  A ,  y ) ) ) ) )  ->  F  =  G )
 
Theoremfrrlem1 25584* Lemma for founded recursion. The final item we are interested in is the union of acceptable functions  B. This lemma just changes bound variables for later use. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  B  =  {
 g  |  E. z
 ( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w )  =  ( w G ( g  |`  Pred
 ( R ,  A ,  w ) ) ) ) ) }
 
Theoremfrrlem2 25585* Lemma for founded recursion. An acceptable function is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( g  e.  B  ->  Fun  g )
 
Theoremfrrlem3 25586* Lemma for founded recursion. An acceptable function's domain is a subset of  A. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( g  e.  B  ->  dom  g  C_  A )
 
Theoremfrrlem4 25587* Lemma for founded recursion. Properties of the restriction of an acceptable function to the domain of another acceptable function. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( g  |`  ( dom  g  i^i  dom  h ) )  Fn  ( dom  g  i^i  dom  h )  /\  A. a  e.  ( dom  g  i^i 
 dom  h ) ( ( g  |`  ( dom  g  i^i  dom  h ) ) `  a
 )  =  ( a G ( ( g  |`  ( dom  g  i^i 
 dom  h ) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
 
Theoremfrrlem5 25588* Lemma for founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( x g u 
 /\  x h v )  ->  u  =  v ) )
 
Theoremfrrlem5b 25589* Lemma for founded recursion. The union of a subclass of  B is a relationship. (Contributed by Paul Chapman, 29-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( C  C_  B  ->  Rel  U. C )
 
Theoremfrrlem5c 25590* Lemma for founded recursion. The union of a subclass of  B is a function. (Contributed by Paul Chapman, 29-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( C  C_  B  ->  Fun  U. C )
 
Theoremfrrlem5d 25591* Lemma for founded recursion. The domain of the union of a subset of  B is a subset of  A. (Contributed by Paul Chapman, 29-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( C  C_  B  ->  dom  U. C  C_  A )
 
Theoremfrrlem5e 25592* Lemma for founded recursion. The domain of the union of a subset of  B is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   =>    |-  ( C  C_  B  ->  ( X  e.  dom  U. C  ->  Pred ( R ,  A ,  X )  C_  dom  U. C ) )
 
Theoremfrrlem6 25593* Lemma for founded recursion. The union of all acceptable functions is a relationship. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 Rel  F
 
Theoremfrrlem7 25594* Lemma for founded recursion. The domain of  F is a subclass of  A. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 dom  F  C_  A
 
Theoremfrrlem10 25595* Lemma for founded recursion. The union of all acceptable functions is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 Fun  F
 
Theoremfrrlem11 25596* Lemma for founded recursion. Here, we calculate the value of  F (the union of all acceptable functions). (Contributed by Paul Chapman, 21-Apr-2012.)
 |-  R  Fr  A   &    |-  R Se  A   &    |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
 C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  /\  A. y  e.  x  ( f `  y )  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( y  e.  dom  F 
 ->  ( F `  y
 )  =  ( y G ( F  |`  Pred ( R ,  A ,  y ) ) ) )
 
19.7.30  Surreal Numbers
 
Syntaxcsur 25597 Declare the class of all surreal numbers (see df-no 25600).
 class  No
 
Syntaxcslt 25598 Declare the less than relationship over surreal numbers (see df-slt 25601).
 class  < s
 
Syntaxcbday 25599 Declare the birthday function for surreal numbers (see df-bday 25602).
 class  bday
 
Definitiondf-no 25600* Define the class of surreal numbers. The surreal numbers are a proper class of numbers developed by John H. Conway and introduced by Donald Knuth in 1975. They form a proper class into which all ordered fields can be embedded. The approach we take to defining them was first introduced by Hary Goshnor, and is based on the conception of a "sign expansion" of a surreal number. We define the surreals as ordinal-indexed sequences of 
1o and  2o, analagous to Goshnor's  (  -  ) and  (  +  ).

After introducing this definition, we will abstract away from it using axioms that Norman Alling developed in "Foundations of Analysis over Surreal Number Fields." This is done in an effort to be agnostic towards the exact implementation of surreals. (Contributed by Scott Fenton, 9-Jun-2011.)

 |-  No  =  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
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