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Definition df-frec 6112
Description: Define a recursive definition generator on  om (the class of finite ordinals) with characteristic function  F and initial value  I. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our frec operation (especially when df-recs 6026 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple; see frec0g 6118 and frecsuc 6128.

Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4394. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 6129, this definition and df-irdg 6091 restricted to  om produce the same result.

Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.)

Assertion
Ref Expression
df-frec  |- frec ( F ,  I )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) ) } ) )  |`  om )
Distinct variable groups:    x, g, m, F    x, I, g, m

Detailed syntax breakdown of Definition df-frec
StepHypRef Expression
1 cF . . 3  class  F
2 cI . . 3  class  I
31, 2cfrec 6111 . 2  class frec ( F ,  I )
4 vg . . . . 5  setvar  g
5 cvv 2615 . . . . 5  class  _V
64cv 1286 . . . . . . . . . . 11  class  g
76cdm 4413 . . . . . . . . . 10  class  dom  g
8 vm . . . . . . . . . . . 12  setvar  m
98cv 1286 . . . . . . . . . . 11  class  m
109csuc 4168 . . . . . . . . . 10  class  suc  m
117, 10wceq 1287 . . . . . . . . 9  wff  dom  g  =  suc  m
12 vx . . . . . . . . . . 11  setvar  x
1312cv 1286 . . . . . . . . . 10  class  x
149, 6cfv 4983 . . . . . . . . . . 11  class  ( g `
 m )
1514, 1cfv 4983 . . . . . . . . . 10  class  ( F `
 ( g `  m ) )
1613, 15wcel 1436 . . . . . . . . 9  wff  x  e.  ( F `  (
g `  m )
)
1711, 16wa 102 . . . . . . . 8  wff  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `  m ) ) )
18 com 4380 . . . . . . . 8  class  om
1917, 8, 18wrex 2356 . . . . . . 7  wff  E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )
20 c0 3275 . . . . . . . . 9  class  (/)
217, 20wceq 1287 . . . . . . . 8  wff  dom  g  =  (/)
2213, 2wcel 1436 . . . . . . . 8  wff  x  e.  I
2321, 22wa 102 . . . . . . 7  wff  ( dom  g  =  (/)  /\  x  e.  I )
2419, 23wo 662 . . . . . 6  wff  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) )
2524, 12cab 2071 . . . . 5  class  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) ) }
264, 5, 25cmpt 3876 . . . 4  class  ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) ) } )
2726crecs 6025 . . 3  class recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) ) } ) )
2827, 18cres 4415 . 2  class  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) ) } ) )  |`  om )
293, 28wceq 1287 1  wff frec ( F ,  I )  =  (recs ( ( g  e.  _V  |->  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) ) } ) )  |`  om )
Colors of variables: wff set class
This definition is referenced by:  freceq1  6113  freceq2  6114  frecex  6115  frecfun  6116  nffrec  6117  frec0g  6118  frecfnom  6122  freccllem  6123  frecfcllem  6125  frecsuclem  6127
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