Definition df-frec 6281

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 6195 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 6287 and frecsuc 6297.

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 4513. 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 6298, this definition and df-irdg 6260 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 6280	. 2 class frec ( F , I )
4		vg	. . . . 5 setvar g
5		cvv 2681	. . . . 5 class _V
6	4	cv 1330	. . . . . . . . . . 11 class g
7	6	cdm 4534	. . . . . . . . . 10 class dom g
8		vm	. . . . . . . . . . . 12 setvar m
9	8	cv 1330	. . . . . . . . . . 11 class m
10	9	csuc 4282	. . . . . . . . . 10 class suc m
11	7, 10	wceq 1331	. . . . . . . . 9 wff dom g = suc m
12		vx	. . . . . . . . . . 11 setvar x
13	12	cv 1330	. . . . . . . . . 10 class x
14	9, 6	cfv 5118	. . . . . . . . . . 11 class ( g ` m )
15	14, 1	cfv 5118	. . . . . . . . . 10 class ( F ` ( g ` m ) )
16	13, 15	wcel 1480	. . . . . . . . 9 wff x e. ( F ` ( g ` m ) )
17	11, 16	wa 103	. . . . . . . 8 wff ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
18		com 4499	. . . . . . . 8 class om
19	17, 8, 18	wrex 2415	. . . . . . 7 wff E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
20		c0 3358	. . . . . . . . 9 class (/)
21	7, 20	wceq 1331	. . . . . . . 8 wff dom g = (/)
22	13, 2	wcel 1480	. . . . . . . 8 wff x e. I
23	21, 22	wa 103	. . . . . . 7 wff ( dom g = (/) /\ x e. I )
24	19, 23	wo 697	. . . . . 6 wff ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) )
25	24, 12	cab 2123	. . . . 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 3984	. . . 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 6194	. . 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 4536	. 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 1331	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  6282  freceq2  6283  frecex  6284  frecfun  6285  nffrec  6286  frec0g  6287  frecfnom  6291  freccllem  6292  frecfcllem  6294  frecsuclem  6296