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Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempreduz 23601 The value of the predecessor class over an upper integer set. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  Pred (  <  ,  ( ZZ>= `  M ) ,  N )  =  ( M ... ( N  -  1 ) ) )
 
Theoremprednn 23602 The value of the predecessor class over the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
 |-  ( N  e.  NN  ->  Pred
 (  <  ,  NN ,  N )  =  ( 1 ... ( N  -  1 ) ) )
 
Theoremprednn0 23603 The value of the predecessor class over  NN0. (Contributed by Scott Fenton, 9-May-2014.)
 |-  ( N  e.  NN0  ->  Pred (  <  ,  NN0 ,  N )  =  ( 0 ... ( N  -  1
 ) ) )
 
Theorempredfz 23604 Calculate the predecessor of an integer under a finite set of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  ( K  e.  ( M ... N )  ->  Pred (  <  ,  ( M ... N ) ,  K )  =  ( M ... ( K  -  1
 ) ) )
 
18.7.14  (Trans)finite Recursion Theorems
 
Theoremtfisg 23605* A closed form of tfis 4644. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( A. x  e.  On  ( A. y  e.  x  [. y  /  x ]. ph 
 ->  ph )  ->  A. x  e.  On  ph )
 
18.7.15  Well-founded induction
 
Theoremtz6.26 23606* All nonempty (possibly proper) subclasses of  A, which has a well-founded relation  R, have  R-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  We  A  /\  R Se  A ) 
 /\  ( B  C_  A  /\  B  =/=  (/) ) ) 
 ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
 
Theoremtz6.26i 23607* All nonempty (possibly proper) subclasses of  A, which has a well-founded relation  R, have  R-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   =>    |-  ( ( B 
 C_  A  /\  B  =/= 
 (/) )  ->  E. y  e.  B  Pred ( R ,  B ,  y )  =  (/) )
 
Theoremwfi 23608* The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if  B is a subclass of a well-ordered class  A with the property that every element of  B whose inital segment is included in 
A is itself equal to  A. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  (
 ( ( R  We  A  /\  R Se  A ) 
 /\  ( B  C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) ) )  ->  A  =  B )
 
Theoremwfii 23609* The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if  B is a subclass of a well-ordered class  A with the property that every element of  B whose inital segment is included in 
A is itself equal to  A. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  R  We  A   &    |-  R Se  A   =>    |-  ( ( B 
 C_  A  /\  A. y  e.  A  ( Pred ( R ,  A ,  y )  C_  B  ->  y  e.  B ) )  ->  A  =  B )
 
Theoremwfisg 23610* Well-Founded Induction Schema. If a property passes from all elements less than  y of a well founded class  A to  y itself (induction hypothesis), then the property holds for all elements of  A. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) [. z  /  y ]. ph  ->  ph ) )   =>    |-  ( ( R  We  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremwfis 23611* Well-Founded Induction Schema. If all elements less than a given set  x of the well founded class  A have a property (induction hypothesis), then all elements of  A have that property. (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) [. z  /  y ]. ph  ->  ph ) )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremwfis2fg 23612* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  F/ y ps   &    |-  ( y  =  z  ->  ( ph  <->  ps ) )   &    |-  ( y  e.  A  ->  ( A. z  e.  Pred  ( R ,  A ,  y
 ) ps  ->  ph )
 )   =>    |-  ( ( R  We  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremwfis2f 23613* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  R  We  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 )
 
Theoremwfis2g 23614* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
 |-  (
 y  =  z  ->  ( ph  <->  ps ) )   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   =>    |-  ( ( R  We  A  /\  R Se  A ) 
 ->  A. y  e.  A  ph )
 
Theoremwfis2 23615* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  R  We  A   &    |-  R Se  A   &    |-  (
 y  =  z  ->  ( ph  <->  ps ) )   &    |-  (
 y  e.  A  ->  (
 A. z  e.  Pred  ( R ,  A ,  y ) ps  ->  ph ) )   =>    |-  ( y  e.  A  -> 
 ph )
 
Theoremwfis3 23616* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
 |-  R  We  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 )
 
Theoremuzsinds 23617* Strong (or "total") induction principle over a set of upper integers. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  ( ZZ>= `  M )  ->  ( A. y  e.  ( M ... ( x  -  1
 ) ) ps  ->  ph ) )   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  ch )
 
Theoremnnsinds 23618* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  NN  ->  (
 A. y  e.  (
 1 ... ( x  -  1 ) ) ps 
 ->  ph ) )   =>    |-  ( N  e.  NN  ->  ch )
 
Theoremnn0sinds 23619* Strong (or "total") induction principle over the non-negative integers. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  N  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  NN0  ->  ( A. y  e.  (
 0 ... ( x  -  1 ) ) ps 
 ->  ph ) )   =>    |-  ( N  e.  NN0 
 ->  ch )
 
Theoremomsinds 23620* Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  om  ->  ( A. y  e.  x  ps  ->  ph ) )   =>    |-  ( A  e.  om 
 ->  ch )
 
18.7.16  Transitive closure under a relationship
 
Syntaxctrpred 23621 Define the transitive predecessor class as a class.
 class  TrPred ( R ,  A ,  X )
 
Definitiondf-trpred 23622* Define the transitive predecessors of a class  X under a relationship  R and a class  A. This class can be thought of as the "smallest" class containing all elements of  A that are linked to  X by a chain of  R relationships (see trpredtr 23634 and trpredmintr 23635). Definition based off of Lemma 4.2 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory (check The Internet Archive for it now as Prof. Monk appears to have rewritten his website). (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  TrPred ( R ,  A ,  X )  =  U. ran  ( rec ( ( a  e. 
 _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) , 
 Pred ( R ,  A ,  X )
 )  |`  om )
 
Theoremdftrpred2 23623* A definition of the transitive predecessors of a class in terms of indexed union. (Contributed by Scott Fenton, 28-Apr-2012.)
 |-  TrPred ( R ,  A ,  X )  =  U_ i  e. 
 om  ( ( rec ( ( a  e. 
 _V  |->  U_ y  e.  a  Pred ( R ,  A ,  y ) ) , 
 Pred ( R ,  A ,  X )
 )  |`  om ) `  i )
 
Theoremtrpredeq1 23624 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( R  =  S  ->  TrPred ( R ,  A ,  X )  =  TrPred ( S ,  A ,  X ) )
 
Theoremtrpredeq2 23625 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  =  B  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  B ,  X ) )
 
Theoremtrpredeq3 23626 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( X  =  Y  ->  TrPred ( R ,  A ,  X )  =  TrPred ( R ,  A ,  Y ) )
 
Theoremtrpredeq1d 23627 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 23628 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 23629 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 23630* 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 23632 and trpredmintr 23635. (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 23631* 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 23632 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 23633 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 23634 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 23635* 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 23636 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 23637* 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 23638* 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 23639 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 23640 The class of transitive predecessors is empty when  A is empty. (Contributed by Scott Fenton, 30-Apr-2012.)
 |-  TrPred ( R ,  (/) ,  X )  =  (/)
 
Theoremtrpredex 23641 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 23642* 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 ) ) )
 
18.7.17  Founded Induction
 
Theoremfrmin 23643* 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 23606 and tz7.5 4412. (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 23644* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 23643). 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 is itself equal to  A. Compare wfi 23608 and tfi 4643, 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 23645* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 23643). 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 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 23646* 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 23647* 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 23648* 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 23649* 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 23650* 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 23651* 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 23652* 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 )
 
18.7.18  Ordering Ordinal Sequences
 
Theoremorderseqlem 23653* Lemma for poseq 23654 and soseq 23655. 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 23654* 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 23655* 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
 
18.7.19  Well-founded recursion
 
Theoremwfr3g 23656* 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 23657* 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
 )  =  ( 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 )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w ) ) ) ) }
 
Theoremwfrlem2 23658* 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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( g  e.  B  ->  Fun  g )
 
Theoremwfrlem3 23659* 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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( g  e.  B  ->  dom  g  C_  A )
 
Theoremwfrlem4 23660* 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 )  =  ( 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
 )  =  ( G `
  ( ( g  |`  ( dom  g  i^i 
 dom  h ) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
 
Theoremwfrlem5 23661* 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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   =>    |-  ( ( g  e.  B  /\  h  e.  B )  ->  (
 ( x g u 
 /\  x h v )  ->  u  =  v ) )
 
Theoremwfrlem6 23662* Lemma for well-founded recursion. The union of all acceptable functions is a relationship. (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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 Rel  F
 
Theoremwfrlem7 23663* Lemma for well-founded recursion. The domain of  F is a subclass 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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 dom  F  C_  A
 
Theoremwfrlem8 23664* 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.)
 |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( Pred ( R ,  ( A  \  dom  F ) ,  X )  =  (/)  <->  Pred ( R ,  A ,  X )  =  Pred ( R ,  dom  F ,  X ) )
 
Theoremwfrlem9 23665* 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.)
 |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( X  e.  dom  F 
 ->  Pred ( R ,  A ,  X )  C_ 
 dom  F )
 
Theoremwfrlem10 23666* 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   &    |-  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 )  =  ( G `  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( ( z  e.  ( A  \  dom  F )  /\  Pred ( R ,  ( A  \ 
 dom  F ) ,  z
 )  =  (/) )  ->  Pred ( R ,  A ,  z )  =  dom  F )
 
Theoremwfrlem11 23667* 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   &    |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |- 
 Fun  F
 
Theoremwfrlem12 23668* Lemma for well-founded recursion. Here, we compute the value of  F (the union of all acceptable functions). (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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( y  e.  dom  F 
 ->  ( F `  y
 )  =  ( G `
  ( F  |`  Pred ( R ,  A ,  y ) ) ) )
 
Theoremwfrlem13 23669* Lemma for well-founded recursion. From here through wfrlem16 23672, 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   &    |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
 ) ) >. } )   =>    |-  (
 z  e.  ( A 
 \  dom  F )  ->  C  Fn  ( dom 
 F  u.  { z } ) )
 
Theoremwfrlem14 23670* Lemma for well-founded recursion. Compute the value of  C. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   &    |-  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 23671* 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   &    |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
 ) ) >. } )   =>    |-  (
 ( z  e.  ( A  \  dom  F ) 
 /\  Pred ( R ,  ( A  \  dom  F ) ,  z )  =  (/) )  ->  C  e.  B )
 
Theoremwfrlem16 23672* 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 23663),  dom  F  =  A. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   &    |-  C  =  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
 ) ) >. } )   =>    |-  dom  F  =  A
 
Theoremwfr1 23673* 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   &    |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  F  Fn  A
 
Theoremwfr2 23674* 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   &    |-  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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( z  e.  A  ->  ( F `  z
 )  =  ( G `
  ( F  |`  Pred ( R ,  A ,  z ) ) ) )
 
Theoremwfr2c 23675* Generalize wfr2 23674 to class arguments. (Contributed by Scott Fenton, 6-Aug-2013.) (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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( X  e.  A  ->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X ) ) ) )
 
Theoremwfr3 23676* 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 23673 and wfr2 23674 is identical to  F. (Contributed by Scott Fenton, 18-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
 )  =  ( G `
  ( f  |`  Pred
 ( R ,  A ,  y ) ) ) ) }   &    |-  F  =  U. B   =>    |-  ( ( H  Fn  A  /\  A. z  e.  A  ( H `  z )  =  ( G `  ( H  |`  Pred ( R ,  A ,  z ) ) ) )  ->  F  =  H )
 
18.7.20  Transfinite recursion via Well-founded recursion
 
TheoremtfrALTlem 23677* Lemma for deriving transfinite recursion from well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.)
 |-  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  y ) ) ) }  =  { f  |  E. x ( f  Fn  x  /\  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x ) 
 /\  A. y  e.  x  ( f `  y
 )  =  ( G `
  ( f  |`  Pred
 (  _E  ,  On ,  y ) ) ) ) }
 
Theoremtfr1ALT 23678* tfr1 6408 via well-founded recursion. (Contributed by Scott Fenton, 17-Aug-1994.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  A  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  F  =  U. A   =>    |-  F  Fn  On
 
Theoremtfr2ALT 23679* tfr2 6409 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  A  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  F  =  U. A   =>    |-  ( z  e.  On  ->  ( F `  z
 )  =  ( G `
  ( F  |`  z ) ) )
 
Theoremtfr3ALT 23680* tfr3 6410 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)
 |-  A  =  { f  |  E. x  e.  On  (
 f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }   &    |-  F  =  U. A   =>    |-  ( ( B  Fn  On  /\  A. x  e. 
 On  ( B `  x )  =  ( G `  ( B  |`  x ) ) )  ->  B  =  F )
 
18.7.21  Founded Recursion
 
Theoremfrr3g 23681* 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 23682* 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 23683* 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 23684* 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 23685* 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 23686* 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 23687* 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 23688* 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 23689* 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 23690* 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 23691* 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 23692* 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 23693* 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 23694* 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 ) ) ) )
 
18.7.22  Surreal Numbers
 
Syntaxcsur 23695 Declare the class of all surreal numbers (see df-no 23698).
 class  No
 
Syntaxcslt 23696 Declare the less than relationship over surreal numbers (see df-slt 23699).
 class  < s
 
Syntaxcbday 23697 Declare the birthday function for surreal numbers (see df-bday 23700).
 class  bday
 
Definitiondf-no 23698* 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 a 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 } }
 
Definitiondf-slt 23699* Next, we introduce surreal less-than, a comparison relationship over the surreals by lexicographically ordering them. (Contributed by Scott Fenton, 9-Jun-2011.)
 |-  < s  =  { <. f ,  g >.  |  (
 ( f  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
 )  =  ( g `
  y )  /\  ( f `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `
  x ) ) ) }
 
Definitiondf-bday 23700 Finally, we introduce the birthday function. This function maps each surreal to an ordinal. In our implementation, this is the domain of the sign function. The important properties of this function are established later. (Contributed by Scott Fenton, 11-Jun-2011.)
 |-  bday  =  ( x  e.  No  |->  dom  x )
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