Definition df-frec 6667

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 6581 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 6673 and frecsuc 6683.

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 4753. 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 6684, this definition and df-irdg 6646 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 6666	. 2 class frec(𝐹, 𝐼)
4		vg	. . . . 5 setvar 𝑔
5		cvv 2821	. . . . 5 class V
6	4	cv 1401	. . . . . . . . . . 11 class 𝑔
7	6	cdm 4776	. . . . . . . . . 10 class dom 𝑔
8		vm	. . . . . . . . . . . 12 setvar 𝑚
9	8	cv 1401	. . . . . . . . . . 11 class 𝑚
10	9	csuc 4512	. . . . . . . . . 10 class suc 𝑚
11	7, 10	wceq 1402	. . . . . . . . 9 wff dom 𝑔 = suc 𝑚
12		vx	. . . . . . . . . . 11 setvar 𝑥
13	12	cv 1401	. . . . . . . . . 10 class 𝑥
14	9, 6	cfv 5380	. . . . . . . . . . 11 class (𝑔‘𝑚)
15	14, 1	cfv 5380	. . . . . . . . . 10 class (𝐹‘(𝑔‘𝑚))
16	13, 15	wcel 2209	. . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔‘𝑚))
17	11, 16	wa 104	. . . . . . . 8 wff (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
18		com 4739	. . . . . . . 8 class ω
19	17, 8, 18	wrex 2529	. . . . . . 7 wff ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
20		c0 3520	. . . . . . . . 9 class ∅
21	7, 20	wceq 1402	. . . . . . . 8 wff dom 𝑔 = ∅
22	13, 2	wcel 2209	. . . . . . . 8 wff 𝑥 ∈ 𝐼
23	21, 22	wa 104	. . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼)
24	19, 23	wo 720	. . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))
25	24, 12	cab 2224	. . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}
26	4, 5, 25	cmpt 4193	. . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})
27	26	crecs 6580	. . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}))
28	27, 18	cres 4778	. 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 1402	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 used by:  freceq1  6668  freceq2  6669  frecex  6670  frecfun  6671  nffrec  6672  frec0g  6673  frecfnom  6677  freccllem  6678  frecfcllem  6680  frecsuclem  6682