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Definition df-frec 6392
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 6306 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 6398 and frecsuc 6408.

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 4604. 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 6409, this definition and df-irdg 6371 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 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)
Distinct variable groups:   𝑥,𝑔,𝑚,𝐹   𝑥,𝐼,𝑔,𝑚

Detailed syntax breakdown of Definition df-frec
StepHypRef Expression
1 cF . . 3 class 𝐹
2 cI . . 3 class 𝐼
31, 2cfrec 6391 . 2 class frec(𝐹, 𝐼)
4 vg . . . . 5 setvar 𝑔
5 cvv 2738 . . . . 5 class V
64cv 1352 . . . . . . . . . . 11 class 𝑔
76cdm 4627 . . . . . . . . . 10 class dom 𝑔
8 vm . . . . . . . . . . . 12 setvar 𝑚
98cv 1352 . . . . . . . . . . 11 class 𝑚
109csuc 4366 . . . . . . . . . 10 class suc 𝑚
117, 10wceq 1353 . . . . . . . . 9 wff dom 𝑔 = suc 𝑚
12 vx . . . . . . . . . . 11 setvar 𝑥
1312cv 1352 . . . . . . . . . 10 class 𝑥
149, 6cfv 5217 . . . . . . . . . . 11 class (𝑔𝑚)
1514, 1cfv 5217 . . . . . . . . . 10 class (𝐹‘(𝑔𝑚))
1613, 15wcel 2148 . . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔𝑚))
1711, 16wa 104 . . . . . . . 8 wff (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))
18 com 4590 . . . . . . . 8 class ω
1917, 8, 18wrex 2456 . . . . . . 7 wff 𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))
20 c0 3423 . . . . . . . . 9 class
217, 20wceq 1353 . . . . . . . 8 wff dom 𝑔 = ∅
2213, 2wcel 2148 . . . . . . . 8 wff 𝑥𝐼
2321, 22wa 104 . . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥𝐼)
2419, 23wo 708 . . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))
2524, 12cab 2163 . . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))}
264, 5, 25cmpt 4065 . . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})
2726crecs 6305 . . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))}))
2827, 18cres 4629 . 2 class (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)
293, 28wceq 1353 1 wff frec(𝐹, 𝐼) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)
Colors of variables: wff set class
This definition is referenced by:  freceq1  6393  freceq2  6394  frecex  6395  frecfun  6396  nffrec  6397  frec0g  6398  frecfnom  6402  freccllem  6403  frecfcllem  6405  frecsuclem  6407
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