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Definition df-irdg 6041
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 5976 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 6062 and for suitable characteristic functions df-frec 6062 yields the same result as  rec restricted to  om, as seen at frecrdg 6079.

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 6040 . 2  class  rec ( F ,  I )
4 vg . . . 4  setvar  g
5 cvv 2611 . . . 4  class  _V
6 vx . . . . . 6  setvar  x
74cv 1284 . . . . . . 7  class  g
87cdm 4392 . . . . . 6  class  dom  g
96cv 1284 . . . . . . . 8  class  x
109, 7cfv 4953 . . . . . . 7  class  ( g `
 x )
1110, 1cfv 4953 . . . . . 6  class  ( F `
 ( g `  x ) )
126, 8, 11ciun 3699 . . . . 5  class  U_ x  e.  dom  g ( F `
 ( g `  x ) )
132, 12cun 2981 . . . 4  class  ( I  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) )
144, 5, 13cmpt 3860 . . 3  class  ( g  e.  _V  |->  ( I  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) ) )
1514crecs 5975 . 2  class recs ( ( g  e.  _V  |->  ( I  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) )
163, 15wceq 1285 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  6042  rdgeq2  6043  rdgfun  6044  rdgexggg  6048  rdgifnon  6050  rdgifnon2  6051  rdgivallem  6052  rdgon  6057  rdg0  6058
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