Definition df-frec 6332

Description: Define a recursive definition generator on ω (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 6246 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 6338 and frecsuc 6348.

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 4561. 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 6349, this definition and df-irdg 6311 restricted to ω 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 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) ↾ ω)

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 6331	. 2 class frec(𝐹, 𝐼)
4		vg	. . . . 5 setvar 𝑔
5		cvv 2712	. . . . 5 class V
6	4	cv 1334	. . . . . . . . . . 11 class 𝑔
7	6	cdm 4583	. . . . . . . . . 10 class dom 𝑔
8		vm	. . . . . . . . . . . 12 setvar 𝑚
9	8	cv 1334	. . . . . . . . . . 11 class 𝑚
10	9	csuc 4324	. . . . . . . . . 10 class suc 𝑚
11	7, 10	wceq 1335	. . . . . . . . 9 wff dom 𝑔 = suc 𝑚
12		vx	. . . . . . . . . . 11 setvar 𝑥
13	12	cv 1334	. . . . . . . . . 10 class 𝑥
14	9, 6	cfv 5167	. . . . . . . . . . 11 class (𝑔‘𝑚)
15	14, 1	cfv 5167	. . . . . . . . . 10 class (𝐹‘(𝑔‘𝑚))
16	13, 15	wcel 2128	. . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔‘𝑚))
17	11, 16	wa 103	. . . . . . . 8 wff (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
18		com 4547	. . . . . . . 8 class ω
19	17, 8, 18	wrex 2436	. . . . . . 7 wff ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
20		c0 3394	. . . . . . . . 9 class ∅
21	7, 20	wceq 1335	. . . . . . . 8 wff dom 𝑔 = ∅
22	13, 2	wcel 2128	. . . . . . . 8 wff 𝑥 ∈ 𝐼
23	21, 22	wa 103	. . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼)
24	19, 23	wo 698	. . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))
25	24, 12	cab 2143	. . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}
26	4, 5, 25	cmpt 4025	. . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})
27	26	crecs 6245	. . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}))
28	27, 18	cres 4585	. 2 class (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) ↾ ω)
29	3, 28	wceq 1335	1 wff frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})) ↾ ω)

This definition is referenced by:  freceq1  6333  freceq2  6334  frecex  6335  frecfun  6336  nffrec  6337  frec0g  6338  frecfnom  6342  freccllem  6343  frecfcllem  6345  frecsuclem  6347