Definition df-frec 6450

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 6364 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 6456 and frecsuc 6466.

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 4641. 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 6467, this definition and df-irdg 6429 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 6449	. 2 class frec(𝐹, 𝐼)
4		vg	. . . . 5 setvar 𝑔
5		cvv 2763	. . . . 5 class V
6	4	cv 1363	. . . . . . . . . . 11 class 𝑔
7	6	cdm 4664	. . . . . . . . . 10 class dom 𝑔
8		vm	. . . . . . . . . . . 12 setvar 𝑚
9	8	cv 1363	. . . . . . . . . . 11 class 𝑚
10	9	csuc 4401	. . . . . . . . . 10 class suc 𝑚
11	7, 10	wceq 1364	. . . . . . . . 9 wff dom 𝑔 = suc 𝑚
12		vx	. . . . . . . . . . 11 setvar 𝑥
13	12	cv 1363	. . . . . . . . . 10 class 𝑥
14	9, 6	cfv 5259	. . . . . . . . . . 11 class (𝑔‘𝑚)
15	14, 1	cfv 5259	. . . . . . . . . 10 class (𝐹‘(𝑔‘𝑚))
16	13, 15	wcel 2167	. . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔‘𝑚))
17	11, 16	wa 104	. . . . . . . 8 wff (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
18		com 4627	. . . . . . . 8 class ω
19	17, 8, 18	wrex 2476	. . . . . . 7 wff ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
20		c0 3451	. . . . . . . . 9 class ∅
21	7, 20	wceq 1364	. . . . . . . 8 wff dom 𝑔 = ∅
22	13, 2	wcel 2167	. . . . . . . 8 wff 𝑥 ∈ 𝐼
23	21, 22	wa 104	. . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼)
24	19, 23	wo 709	. . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))
25	24, 12	cab 2182	. . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}
26	4, 5, 25	cmpt 4095	. . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})
27	26	crecs 6363	. . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}))
28	27, 18	cres 4666	. 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 1364	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  6451  freceq2  6452  frecex  6453  frecfun  6454  nffrec  6455  frec0g  6456  frecfnom  6460  freccllem  6461  frecfcllem  6463  frecsuclem  6465