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Definition df-irdg 6089
Description: Define a recursive definition generator on  On (the class of ordinal numbers) 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 
rec operation (especially when df-recs 6024 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple. In classical logic it would be easier to divide this definition into cases based on whether the domain of  g is zero, a successor, or a limit ordinal. Cases do not (in general) work that way in intuitionistic logic, so instead we choose a definition which takes the union of all the results of the characteristic function for ordinals in the domain of  g. This means that this definition has the expected properties for increasing and continuous ordinal functions, which include ordinal addition and multiplication.

For finite recursion we also define df-frec 6110 and for suitable characteristic functions df-frec 6110 yields the same result as  rec restricted to  om, as seen at frecrdg 6127.

Note: We introduce 
rec 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 Jim Kingdon, 19-May-2019.)

Assertion
Ref Expression
df-irdg  |-  rec ( F ,  I )  = recs ( ( g  e. 
_V  |->  ( I  u. 
U_ x  e.  dom  g ( F `  ( g `  x
) ) ) ) )
Distinct variable groups:    x, g, F   
x, I, g

Detailed syntax breakdown of Definition df-irdg
StepHypRef Expression
1 cF . . 3  class  F
2 cI . . 3  class  I
31, 2crdg 6088 . 2  class  rec ( F ,  I )
4 vg . . . 4  setvar  g
5 cvv 2615 . . . 4  class  _V
6 vx . . . . . 6  setvar  x
74cv 1286 . . . . . . 7  class  g
87cdm 4411 . . . . . 6  class  dom  g
96cv 1286 . . . . . . . 8  class  x
109, 7cfv 4981 . . . . . . 7  class  ( g `
 x )
1110, 1cfv 4981 . . . . . 6  class  ( F `
 ( g `  x ) )
126, 8, 11ciun 3713 . . . . 5  class  U_ x  e.  dom  g ( F `
 ( g `  x ) )
132, 12cun 2986 . . . 4  class  ( I  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )
144, 5, 13cmpt 3874 . . 3  class  ( g  e.  _V  |->  ( I  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) ) )
1514crecs 6023 . 2  class recs ( ( g  e.  _V  |->  ( I  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) )
163, 15wceq 1287 1  wff  rec ( F ,  I )  = recs ( ( g  e. 
_V  |->  ( I  u. 
U_ x  e.  dom  g ( F `  ( g `  x
) ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  rdgeq1  6090  rdgeq2  6091  rdgfun  6092  rdgexggg  6096  rdgifnon  6098  rdgifnon2  6099  rdgivallem  6100  rdgon  6105  rdg0  6106
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