Definition df-frec 6477

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 6391 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 6483 and frecsuc 6493.

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 4652. 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 6494, this definition and df-irdg 6456 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

Step	Hyp	Ref	Expression
1		cF	. . 3 class F
2		cI	. . 3 class I
3	1, 2	cfrec 6476	. 2 class frec ( F , I )
4		vg	. . . . 5 setvar g
5		cvv 2772	. . . . 5 class _V
6	4	cv 1372	. . . . . . . . . . 11 class g
7	6	cdm 4675	. . . . . . . . . 10 class dom g
8		vm	. . . . . . . . . . . 12 setvar m
9	8	cv 1372	. . . . . . . . . . 11 class m
10	9	csuc 4412	. . . . . . . . . 10 class suc m
11	7, 10	wceq 1373	. . . . . . . . 9 wff dom g = suc m
12		vx	. . . . . . . . . . 11 setvar x
13	12	cv 1372	. . . . . . . . . 10 class x
14	9, 6	cfv 5271	. . . . . . . . . . 11 class ( g ` m )
15	14, 1	cfv 5271	. . . . . . . . . 10 class ( F ` ( g ` m ) )
16	13, 15	wcel 2176	. . . . . . . . 9 wff x e. ( F ` ( g ` m ) )
17	11, 16	wa 104	. . . . . . . 8 wff ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
18		com 4638	. . . . . . . 8 class om
19	17, 8, 18	wrex 2485	. . . . . . 7 wff E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
20		c0 3460	. . . . . . . . 9 class (/)
21	7, 20	wceq 1373	. . . . . . . 8 wff dom g = (/)
22	13, 2	wcel 2176	. . . . . . . 8 wff x e. I
23	21, 22	wa 104	. . . . . . 7 wff ( dom g = (/) /\ x e. I )
24	19, 23	wo 710	. . . . . 6 wff ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) )
25	24, 12	cab 2191	. . . . 5 class { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) ) }
26	4, 5, 25	cmpt 4105	. . . 4 class ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) ) } )
27	26	crecs 6390	. . 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 ) ) } ) )
28	27, 18	cres 4677	. 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 )
29	3, 28	wceq 1373	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 )

This definition is referenced by:  freceq1  6478  freceq2  6479  frecex  6480  frecfun  6481  nffrec  6482  frec0g  6483  frecfnom  6487  freccllem  6488  frecfcllem  6490  frecsuclem  6492