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Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtposfn 7001 Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  B )  -> tpos  F  Fn  ( B  X.  A ) )
 
Theoremtpos0 7002 Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
 |- tpos  (/) 
 =  (/)
 
Theoremtposco 7003 Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- tpos 
 ( F  o.  G )  =  ( F  o. tpos  G )
 
Theoremtpossym 7004* Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( F  Fn  ( A  X.  A )  ->  (tpos  F  =  F  <->  A. x  e.  A  A. y  e.  A  ( x F y )  =  ( y F x ) ) )
 
Theoremtposeqi 7005 Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  G   =>    |- tpos  F  = tpos  G
 
Theoremtposex 7006 A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  e.  _V   =>    |- tpos  F  e.  _V
 
Theoremnftpos 7007 Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F/_ x F   =>    |-  F/_ xtpos  F
 
Theoremtposoprab 7008* Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  { <. <. x ,  y >. ,  z >.  |  ph }   =>    |- tpos  F  =  { <.
 <. y ,  x >. ,  z >.  |  ph }
 
Theoremtposmpt2 7009* Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |- tpos  F  =  (
 y  e.  B ,  x  e.  A  |->  C )
 
Theoremtposconst 7010 The transposition of a constant operation using the relation representation. (Contributed by SO, 11-Jul-2018.)
 |- tpos 
 ( ( A  X.  B )  X.  { C } )  =  (
 ( B  X.  A )  X.  { C }
 )
 
2.4.11  Curry and uncurry
 
Syntaxccur 7011 Extend class notation to include the currying function.
 class curry  A
 
Syntaxcunc 7012 Extend class notation to include the uncurrying function.
 class uncurry  A
 
Definitiondf-cur 7013* Define the currying of  F, which splits a function of two arguments into a function of the first argument, producing a function over the second argument. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- curry  F  =  ( x  e.  dom  dom  F  |->  { <. y ,  z >.  |  <. x ,  y >. F z } )
 
Definitiondf-unc 7014* Define the uncurrying of  F, which takes a function producing functions, and transforms it into a two-argument function. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- uncurry  F  =  { <. <. x ,  y >. ,  z >.  |  y ( F `  x ) z }
 
Theoremmpt2curryd 7015* The currying of an operation given in maps-to notation, splitting the operation (function of two arguments) into a function of the first argument, producing a function over the second argument. (Contributed by AV, 27-Oct-2019.)
 |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )   &    |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  C  e.  V )   &    |-  ( ph  ->  Y  =/=  (/) )   =>    |-  ( ph  -> curry  F  =  ( x  e.  X  |->  ( y  e.  Y  |->  C ) ) )
 
Theoremmpt2curryvald 7016* The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.)
 |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )   &    |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  C  e.  V )   &    |-  ( ph  ->  Y  =/=  (/) )   &    |-  ( ph  ->  Y  e.  W )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  (curry  F `
  A )  =  ( y  e.  Y  |->  [_ A  /  x ]_ C ) )
 
Theoremfvmpt2curryd 7017* The value of the value of a curried operation given in maps-to notation is the operation value of the original operation. (Contributed by AV, 27-Oct-2019.)
 |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )   &    |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  C  e.  V )   &    |-  ( ph  ->  Y  e.  W )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   =>    |-  ( ph  ->  (
 (curry  F `  A ) `
  B )  =  ( A F B ) )
 
2.4.12  Undefined values
 
Syntaxcund 7018 Extend class notation with undefined value function.
 class  Undef
 
Definitiondf-undef 7019 Define the undefined value function, whose value at set  s is guaranteed not to be a member of 
s (see pwuninel 7021). (Contributed by NM, 15-Sep-2011.)
 |- 
 Undef  =  ( s  e.  _V  |->  ~P U. s )
 
Theorempwuninel2 7020 Direct proof of pwuninel 7021 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( U. A  e.  V  ->  -.  ~P U. A  e.  A )
 
Theorempwuninel 7021 The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 7020. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |- 
 -.  ~P U. A  e.  A
 
Theoremundefval 7022 Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7024 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( S  e.  V  ->  ( Undef `  S )  =  ~P U. S )
 
Theoremundefnel2 7023 The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)
 |-  ( S  e.  V  ->  -.  ( Undef `  S )  e.  S )
 
Theoremundefnel 7024 The undefined value generated from a set is not a member of the set. (Contributed by NM, 15-Sep-2011.)
 |-  ( S  e.  V  ->  ( Undef `  S )  e/  S )
 
Theoremundefne0 7025 The undefined value generated from a set is not empty. (Contributed by NM, 3-Sep-2018.)
 |-  ( S  e.  V  ->  ( Undef `  S )  =/= 
 (/) )
 
2.4.13  Well-founded recursion
 
Syntaxcwrecs 7026 Declare syntax for the well-founded recursive function generator.
 class wrecs ( R ,  A ,  F )
 
Definitiondf-wrecs 7027* Here we define the well-founded recursive function generator. This function takes the usual expressions from recursion theorems and forms a unified definition. Specifically, given a function  F, a relationship  R, and a base set  A, this definition generates a function  G  = wrecs ( R ,  A ,  F ) that has property that, at any point  x  e.  A,  ( G `  x )  =  ( F `  ( G  |  `  Pred ( R ,  A ,  x ) ) ). See wfr1 7053, wfr2 7054, and wfr3 7055. (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 7028 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 7029 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 7030 Equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
 |-  ( R  =  S  -> wrecs ( R ,  A ,  F )  = wrecs ( S ,  A ,  F ) )
 
Theoremwrecseq2 7031 Equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
 |-  ( A  =  B  -> wrecs ( R ,  A ,  F )  = wrecs ( R ,  B ,  F ) )
 
Theoremwrecseq3 7032 Equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
 |-  ( F  =  G  -> wrecs ( R ,  A ,  F )  = wrecs ( R ,  A ,  G ) )
 
Theoremwfr3g 7033* 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 7034* 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 7035* 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 7036* 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 )
 
Theoremwfrlem3a 7037* Lemma for well-founded recursion. Show membership in the class of acceptable functions. (Contributed by Scott Fenton, 31-Jul-2020.)
 |-  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.  _V   =>    |-  ( G  e.  B  <->  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 ) ) ) ) )
 
Theoremwfrlem4 7038* 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 7039* 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 ) )
 
Theoremwfrrel 7040 The well-founded recursion generator generates a relationship. (Contributed by Scott Fenton, 8-Jun-2018.)
 |-  F  = wrecs ( R ,  A ,  G )   =>    |- 
 Rel  F
 
Theoremwfrdmss 7041 The domain of the well-founded recursion generator is a subclass of  A. (Contributed by Scott Fenton, 21-Apr-2011.)
 |-  F  = wrecs ( R ,  A ,  G )   =>    |- 
 dom  F  C_  A
 
Theoremwfrlem8 7042 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 ) )
 
Theoremwfrdmcl 7043 Given  F  = wrecs ( R ,  A ,  X )  /\  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 7044* 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 )
 
Theoremwfrfun 7045 The well-founded function generator generates 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 7046* 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 7047* Lemma for well-founded recursion. From here through wfrlem16 7050, 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 7048* 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 7049* Lemma for well-founded recursion. When  z is  R minimal,  C is an acceptable function. This step is where the Axiom of Replacement becomes required. (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 7050* 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 wfrdmss 7041),  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
 
Theoremwfrlem17 7051 Without using ax-rep 4529, show that all restrictions of wrecs are sets. (Contributed by Scott Fenton, 31-Jul-2020.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( X  e.  dom  F 
 ->  ( F  |`  Pred ( R ,  A ,  X ) )  e. 
 _V )
 
Theoremwfr2a 7052 A weak version of wfr2 7054 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.)
 |-  R  We  A   &    |-  R Se  A   &    |-  F  = wrecs ( R ,  A ,  G )   =>    |-  ( X  e.  dom  F 
 ->  ( F `  X )  =  ( G `  ( F  |`  Pred ( R ,  A ,  X ) ) ) )
 
Theoremwfr1 7053 The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function  G. 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 7054 The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of  F at any  X  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 7055* 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 7053 and wfr2 7054 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 )
 
2.4.14  Functions on ordinals; strictly monotone ordinal functions
 
Theoremiunon 7056* The indexed union of a set of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  e.  On )
 
TheoremiunonOLD 7057* The indexed union of a set of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x  e.  A  B  e.  On  -> 
 U_ x  e.  A  B  e.  On )
 
Theoremiinon 7058* The nonempty indexed intersection of a class of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( A. x  e.  A  B  e.  On  /\  A  =/=  (/) )  ->  |^|_
 x  e.  A  B  e.  On )
 
Theoremonfununi 7059* A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)
 |-  ( Lim  y  ->  ( F `  y )  =  U_ x  e.  y  ( F `  x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( F `  x ) 
 C_  ( F `  y ) )   =>    |-  ( ( S  e.  T  /\  S  C_ 
 On  /\  S  =/=  (/) )  ->  ( F ` 
 U. S )  = 
 U_ x  e.  S  ( F `  x ) )
 
Theoremonovuni 7060* A variant of onfununi 7059 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  ( Lim  y  ->  ( A F y )  =  U_ x  e.  y  ( A F x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x ) 
 C_  ( A F y ) )   =>    |-  ( ( S  e.  T  /\  S  C_ 
 On  /\  S  =/=  (/) )  ->  ( A F U. S )  = 
 U_ x  e.  S  ( A F x ) )
 
Theoremonoviun 7061* A variant of onovuni 7060 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( Lim  y  ->  ( A F y )  =  U_ x  e.  y  ( A F x ) )   &    |-  (
 ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x ) 
 C_  ( A F y ) )   =>    |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( A F U_ z  e.  K  L )  =  U_ z  e.  K  ( A F L ) )
 
Theoremonnseq 7062* There are no length  om decreasing sequences in the ordinals. See also noinfep 8155 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)
 |-  ( ( F `  (/) )  e.  On  ->  E. x  e.  om  -.  ( F `  suc  x )  e.  ( F `  x ) )
 
Syntaxwsmo 7063 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
 wff  Smo  A
 
Definitiondf-smo 7064* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
 |-  ( Smo  A  <->  ( A : dom  A --> On  /\  Ord  dom  A 
 /\  A. x  e.  dom  A
 A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y ) ) ) )
 
Theoremdfsmo2 7065* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( Smo  F  <->  ( F : dom  F --> On  /\  Ord  dom  F 
 /\  A. x  e.  dom  F
 A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
 
Theoremissmo 7066* Conditions for which  A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
 |-  A : B --> On   &    |-  Ord  B   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  e.  y  ->  ( A `  x )  e.  ( A `  y ) ) )   &    |-  dom 
 A  =  B   =>    |-  Smo  A
 
Theoremissmo2 7067* Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( F : A --> B  ->  ( ( B 
 C_  On  /\  Ord  A  /\  A. x  e.  A  A. y  e.  x  ( F `  y )  e.  ( F `  x ) )  ->  Smo  F ) )
 
Theoremsmoeq 7068 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( A  =  B  ->  ( Smo  A  <->  Smo  B ) )
 
Theoremsmodm 7069 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |-  ( Smo  A  ->  Ord 
 dom  A )
 
Theoremsmores 7070 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( Smo  A  /\  B  e.  dom  A )  ->  Smo  ( A  |`  B ) )
 
Theoremsmores3 7071 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( Smo  ( A  |`  B )  /\  C  e.  ( dom  A  i^i  B )  /\  Ord 
 B )  ->  Smo  ( A  |`  C ) )
 
Theoremsmores2 7072 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
 |-  ( ( Smo  F  /\  Ord  A )  ->  Smo  ( F  |`  A ) )
 
Theoremsmodm2 7073 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
 
Theoremsmofvon2 7074 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( Smo  F  ->  ( F `  B )  e.  On )
 
Theoremiordsmo 7075 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)
 |- 
 Ord  A   =>    |- 
 Smo  (  _I  |`  A )
 
Theoremsmo0 7076 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |- 
 Smo  (/)
 
Theoremsmofvon 7077 If  B is a strictly monotone ordinal function, and  A is in the domain of  B, then the value of the function at 
A is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B )  ->  ( B `  A )  e.  On )
 
Theoremsmoel 7078 If  x is less than  y then a strictly monotone function's value will be strictly less at  x than at  y. (Contributed by Andrew Salmon, 22-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B  /\  C  e.  A ) 
 ->  ( B `  C )  e.  ( B `  A ) )
 
Theoremsmoiun 7079* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
 |-  ( ( Smo  B  /\  A  e.  dom  B )  ->  U_ x  e.  A  ( B `  x ) 
 C_  ( B `  A ) )
 
Theoremsmoiso 7080 If  F is an isomorphism from an ordinal  A onto  B, which is a subset of the ordinals, then 
F is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  B  C_  On )  ->  Smo  F )
 
Theoremsmoel2 7081 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( B  e.  A  /\  C  e.  B ) )  ->  ( F `  C )  e.  ( F `  B ) )
 
Theoremsmo11 7082 A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( F : A
 --> B  /\  Smo  F )  ->  F : A -1-1-> B )
 
Theoremsmoord 7083 A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  e.  D  <->  ( F `  C )  e.  ( F `  D ) ) )
 
Theoremsmoword 7084 A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
 |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  C_  D  <->  ( F `  C )  C_  ( F `
  D ) ) )
 
Theoremsmogt 7085 A strictly monotone ordinal function is greater than or equal to its argument. Exercise 1 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 23-Nov-2011.) (Revised by Mario Carneiro, 28-Feb-2013.)
 |-  ( ( F  Fn  A  /\  Smo  F  /\  C  e.  A )  ->  C  C_  ( F `  C ) )
 
Theoremsmorndom 7086 The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)
 |-  ( ( F : A
 --> B  /\  Smo  F  /\  Ord  B )  ->  A  C_  B )
 
Theoremsmoiso2 7087 The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of  On. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( ( Ord  A  /\  B  C_  On )  ->  ( ( F : A -onto-> B  /\  Smo  F ) 
 <->  F  Isom  _E  ,  _E  ( A ,  B ) ) )
 
2.4.15  "Strong" transfinite recursion
 
Syntaxcrecs 7088 Notation for a function defined by strong transfinite recursion.
 class recs ( F )
 
Definitiondf-recs 7089 Define a function recs ( F ) on  On, the class of ordinal numbers, by transfinite recursion given a rule  F which sets the next value given all values so far. See df-rdg 7127 for more details on why this definition is desirable. Unlike df-rdg 7127 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See recsfnon 7120 and recsval 7121 for the primary contract of this definition. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Scott Fenton, 3-Aug-2020.)
 |- recs
 ( F )  = wrecs
 (  _E  ,  On ,  F )
 
Theoremdfrecs3 7090* The old definition of transfinite recursion. This version is preferred for developement, as it demonstrates the properties of transfinite recursion without relying on well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.)
 |- recs
 ( F )  = 
 U. { f  | 
 E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }
 
Theoremrecseq 7091 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
 
Theoremnfrecs 7092 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F/_ x F   =>    |-  F/_ xrecs ( F )
 
Theoremtfrlem1 7093* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  ( Fun  F  /\  A  C_ 
 dom  F ) )   &    |-  ( ph  ->  ( Fun  G  /\  A  C_  dom  G ) )   &    |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( B `  ( F  |`  x ) ) )   &    |-  ( ph  ->  A. x  e.  A  ( G `  x )  =  ( B `  ( G  |`  x ) ) )   =>    |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
 
Theoremtfrlem3a 7094* Lemma for transfinite recursion. Let  A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in  A for later use. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   &    |-  G  e.  _V   =>    |-  ( G  e.  A  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
 
Theoremtfrlem3 7095* Lemma for transfinite recursion. Let  A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in  A for later use. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  A  =  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( F `  ( g  |`  w ) ) ) }
 
Theoremtfrlem4 7096* Lemma for transfinite recursion.  A is the class of all "acceptable" functions, and  F is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( g  e.  A  ->  Fun  g )
 
Theoremtfrlem5 7097* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |-  ( ( g  e.  A  /\  h  e.  A )  ->  (
 ( x g u 
 /\  x h v )  ->  u  =  v ) )
 
Theoremrecsfval 7098* Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- recs
 ( F )  = 
 U. A
 
Theoremtfrlem6 7099* Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Rel recs ( F )
 
Theoremtfrlem7 7100* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  A  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
 )  =  ( F `
  ( f  |`  y ) ) ) }   =>    |- 
 Fun recs ( F )
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