Definition df-frec 6537

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 6451 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 6543 and frecsuc 6553.

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 4696. 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 6554, this definition and df-irdg 6516 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 6536	. 2 class frec ( F , I )
4		vg	. . . . 5 setvar g
5		cvv 2799	. . . . 5 class _V
6	4	cv 1394	. . . . . . . . . . 11 class g
7	6	cdm 4719	. . . . . . . . . 10 class dom g
8		vm	. . . . . . . . . . . 12 setvar m
9	8	cv 1394	. . . . . . . . . . 11 class m
10	9	csuc 4456	. . . . . . . . . 10 class suc m
11	7, 10	wceq 1395	. . . . . . . . 9 wff dom g = suc m
12		vx	. . . . . . . . . . 11 setvar x
13	12	cv 1394	. . . . . . . . . 10 class x
14	9, 6	cfv 5318	. . . . . . . . . . 11 class ( g ` m )
15	14, 1	cfv 5318	. . . . . . . . . 10 class ( F ` ( g ` m ) )
16	13, 15	wcel 2200	. . . . . . . . 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 4682	. . . . . . . 8 class om
19	17, 8, 18	wrex 2509	. . . . . . 7 wff E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
20		c0 3491	. . . . . . . . 9 class (/)
21	7, 20	wceq 1395	. . . . . . . 8 wff dom g = (/)
22	13, 2	wcel 2200	. . . . . . . 8 wff x e. I
23	21, 22	wa 104	. . . . . . 7 wff ( dom g = (/) /\ x e. I )
24	19, 23	wo 713	. . . . . 6 wff ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) )
25	24, 12	cab 2215	. . . . 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 4145	. . . 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 6450	. . 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 4721	. 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 1395	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  6538  freceq2  6539  frecex  6540  frecfun  6541  nffrec  6542  frec0g  6543  frecfnom  6547  freccllem  6548  frecfcllem  6550  frecsuclem  6552