Definition df-frec 5978

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 5920 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 5983 and frecsuc 5991.

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 4327. 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 5992, this definition and df-irdg 5957 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 5977	. 2 class frec ( F , I )
4		vg	. . . . 5 setvar g
5		cvv 2557	. . . . 5 class _V
6	4	cv 1242	. . . . . . . . . . 11 class g
7	6	cdm 4345	. . . . . . . . . 10 class dom g
8		vm	. . . . . . . . . . . 12 setvar m
9	8	cv 1242	. . . . . . . . . . 11 class m
10	9	csuc 4102	. . . . . . . . . 10 class suc m
11	7, 10	wceq 1243	. . . . . . . . 9 wff dom g = suc m
12		vx	. . . . . . . . . . 11 setvar x
13	12	cv 1242	. . . . . . . . . 10 class x
14	9, 6	cfv 4902	. . . . . . . . . . 11 class ( g ` m )
15	14, 1	cfv 4902	. . . . . . . . . 10 class ( F ` ( g ` m ) )
16	13, 15	wcel 1393	. . . . . . . . 9 wff x e. ( F ` ( g ` m ) )
17	11, 16	wa 97	. . . . . . . 8 wff ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
18		com 4313	. . . . . . . 8 class om
19	17, 8, 18	wrex 2307	. . . . . . 7 wff E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
20		c0 3224	. . . . . . . . 9 class (/)
21	7, 20	wceq 1243	. . . . . . . 8 wff dom g = (/)
22	13, 2	wcel 1393	. . . . . . . 8 wff x e. I
23	21, 22	wa 97	. . . . . . 7 wff ( dom g = (/) /\ x e. I )
24	19, 23	wo 629	. . . . . 6 wff ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) )
25	24, 12	cab 2026	. . . . 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 3818	. . . 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 5919	. . 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 4347	. 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 1243	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  5979  freceq2  5980  frecex  5981  nffrec  5982  frec0g  5983  frecfnom  5986  frecsuclem1  5987  frecsuclem2  5989