Definition df-frec 6391

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 6305 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 6397 and frecsuc 6407.

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 4603. 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 6408, this definition and df-irdg 6370 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 6390	. 2 class frec(𝐹, 𝐼)
4		vg	. . . . 5 setvar 𝑔
5		cvv 2737	. . . . 5 class V
6	4	cv 1352	. . . . . . . . . . 11 class 𝑔
7	6	cdm 4626	. . . . . . . . . 10 class dom 𝑔
8		vm	. . . . . . . . . . . 12 setvar 𝑚
9	8	cv 1352	. . . . . . . . . . 11 class 𝑚
10	9	csuc 4365	. . . . . . . . . 10 class suc 𝑚
11	7, 10	wceq 1353	. . . . . . . . 9 wff dom 𝑔 = suc 𝑚
12		vx	. . . . . . . . . . 11 setvar 𝑥
13	12	cv 1352	. . . . . . . . . 10 class 𝑥
14	9, 6	cfv 5216	. . . . . . . . . . 11 class (𝑔‘𝑚)
15	14, 1	cfv 5216	. . . . . . . . . 10 class (𝐹‘(𝑔‘𝑚))
16	13, 15	wcel 2148	. . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔‘𝑚))
17	11, 16	wa 104	. . . . . . . 8 wff (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
18		com 4589	. . . . . . . 8 class ω
19	17, 8, 18	wrex 2456	. . . . . . 7 wff ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚)))
20		c0 3422	. . . . . . . . 9 class ∅
21	7, 20	wceq 1353	. . . . . . . 8 wff dom 𝑔 = ∅
22	13, 2	wcel 2148	. . . . . . . 8 wff 𝑥 ∈ 𝐼
23	21, 22	wa 104	. . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼)
24	19, 23	wo 708	. . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))
25	24, 12	cab 2163	. . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}
26	4, 5, 25	cmpt 4064	. . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))})
27	26	crecs 6304	. . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐼))}))
28	27, 18	cres 4628	. 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 1353	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  6392  freceq2  6393  frecex  6394  frecfun  6395  nffrec  6396  frec0g  6397  frecfnom  6401  freccllem  6402  frecfcllem  6404  frecsuclem  6406