Definition df-frec 6328

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 6242 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 6334 and frecsuc 6344.

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 4557. 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 6345, this definition and df-irdg 6307 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 6327	. 2 class frec ( F , I )
4		vg	. . . . 5 setvar g
5		cvv 2709	. . . . 5 class _V
6	4	cv 1331	. . . . . . . . . . 11 class g
7	6	cdm 4579	. . . . . . . . . 10 class dom g
8		vm	. . . . . . . . . . . 12 setvar m
9	8	cv 1331	. . . . . . . . . . 11 class m
10	9	csuc 4320	. . . . . . . . . 10 class suc m
11	7, 10	wceq 1332	. . . . . . . . 9 wff dom g = suc m
12		vx	. . . . . . . . . . 11 setvar x
13	12	cv 1331	. . . . . . . . . 10 class x
14	9, 6	cfv 5163	. . . . . . . . . . 11 class ( g ` m )
15	14, 1	cfv 5163	. . . . . . . . . 10 class ( F ` ( g ` m ) )
16	13, 15	wcel 2125	. . . . . . . . 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 4543	. . . . . . . 8 class om
19	17, 8, 18	wrex 2433	. . . . . . 7 wff E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) )
20		c0 3390	. . . . . . . . 9 class (/)
21	7, 20	wceq 1332	. . . . . . . 8 wff dom g = (/)
22	13, 2	wcel 2125	. . . . . . . 8 wff x e. I
23	21, 22	wa 103	. . . . . . 7 wff ( dom g = (/) /\ x e. I )
24	19, 23	wo 698	. . . . . 6 wff ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) )
25	24, 12	cab 2140	. . . . 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 4021	. . . 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 6241	. . 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 4581	. 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 1332	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  6329  freceq2  6330  frecex  6331  frecfun  6332  nffrec  6333  frec0g  6334  frecfnom  6338  freccllem  6339  frecfcllem  6341  frecsuclem  6343