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Definition df-frec 6386
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 6300 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 6392 and frecsuc 6402.

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 4600. 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 6403, this definition and df-irdg 6365 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 6385 . 2  class frec ( F ,  I )
4 vg . . . . 5  setvar  g
5 cvv 2737 . . . . 5  class  _V
64cv 1352 . . . . . . . . . . 11  class  g
76cdm 4623 . . . . . . . . . 10  class  dom  g
8 vm . . . . . . . . . . . 12  setvar  m
98cv 1352 . . . . . . . . . . 11  class  m
109csuc 4362 . . . . . . . . . 10  class  suc  m
117, 10wceq 1353 . . . . . . . . 9  wff  dom  g  =  suc  m
12 vx . . . . . . . . . . 11  setvar  x
1312cv 1352 . . . . . . . . . 10  class  x
149, 6cfv 5212 . . . . . . . . . . 11  class  ( g `
 m )
1514, 1cfv 5212 . . . . . . . . . 10  class  ( F `
 ( g `  m ) )
1613, 15wcel 2148 . . . . . . . . 9  wff  x  e.  ( F `  (
g `  m )
)
1711, 16wa 104 . . . . . . . 8  wff  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `  m ) ) )
18 com 4586 . . . . . . . 8  class  om
1917, 8, 18wrex 2456 . . . . . . 7  wff  E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )
20 c0 3422 . . . . . . . . 9  class  (/)
217, 20wceq 1353 . . . . . . . 8  wff  dom  g  =  (/)
2213, 2wcel 2148 . . . . . . . 8  wff  x  e.  I
2321, 22wa 104 . . . . . . 7  wff  ( dom  g  =  (/)  /\  x  e.  I )
2419, 23wo 708 . . . . . 6  wff  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  ( g `
 m ) ) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) )
2524, 12cab 2163 . . . . 5  class  { x  |  ( E. m  e.  om  ( dom  g  =  suc  m  /\  x  e.  ( F `  (
g `  m )
) )  \/  ( dom  g  =  (/)  /\  x  e.  I ) ) }
264, 5, 25cmpt 4061 . . . 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 6299 . . 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 4625 . 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 1353 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  6387  freceq2  6388  frecex  6389  frecfun  6390  nffrec  6391  frec0g  6392  frecfnom  6396  freccllem  6397  frecfcllem  6399  frecsuclem  6401
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