Definition df-frec 6001

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 5943 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 6006 and frecsuc 6014.

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 4345. 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 6015, this definition and df-irdg 5980 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 6000	. 2 class frec(𝐹, 𝐼)
4		vg	. . . . 5 setvar 𝑔
5		cvv 2601	. . . . 5 class V
6	4	cv 1283	. . . . . . . . . . 11 class 𝑔
7	6	cdm 4363	. . . . . . . . . 10 class dom 𝑔
8		vm	. . . . . . . . . . . 12 setvar 𝑚
9	8	cv 1283	. . . . . . . . . . 11 class 𝑚
10	9	csuc 4120	. . . . . . . . . 10 class suc 𝑚
11	7, 10	wceq 1284	. . . . . . . . 9 wff dom 𝑔 = suc 𝑚
12		vx	. . . . . . . . . . 11 setvar 𝑥
13	12	cv 1283	. . . . . . . . . 10 class 𝑥
14	9, 6	cfv 4922	. . . . . . . . . . 11 class (𝑔‘𝑚)
15	14, 1	cfv 4922	. . . . . . . . . 10 class (𝐹‘(𝑔‘𝑚))
16	13, 15	wcel 1433	. . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔‘𝑚))
17	11, 16	wa 102	. . . . . . . 8 wff (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
18		com 4331	. . . . . . . 8 class ω
19	17, 8, 18	wrex 2349	. . . . . . 7 wff ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
20		c0 3251	. . . . . . . . 9 class ∅
21	7, 20	wceq 1284	. . . . . . . 8 wff dom 𝑔 = ∅
22	13, 2	wcel 1433	. . . . . . . 8 wff 𝑥 ∈ 𝐼
23	21, 22	wa 102	. . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼)
24	19, 23	wo 661	. . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))
25	24, 12	cab 2067	. . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}
26	4, 5, 25	cmpt 3839	. . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})
27	26	crecs 5942	. . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}))
28	27, 18	cres 4365	. 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 1284	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  6002  freceq2  6003  frecex  6004  nffrec  6005  frec0g  6006  frecfnom  6009  frecsuclem1  6010  frecsuclem2  6012