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Definition df-frec 6110
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 6024 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 6116 and frecsuc 6126.

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 4392. 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 6127, this definition and df-irdg 6089 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 6109 . 2 class frec(𝐹, 𝐼)
4 vg . . . . 5 setvar 𝑔
5 cvv 2615 . . . . 5 class V
64cv 1286 . . . . . . . . . . 11 class 𝑔
76cdm 4411 . . . . . . . . . 10 class dom 𝑔
8 vm . . . . . . . . . . . 12 setvar 𝑚
98cv 1286 . . . . . . . . . . 11 class 𝑚
109csuc 4166 . . . . . . . . . 10 class suc 𝑚
117, 10wceq 1287 . . . . . . . . 9 wff dom 𝑔 = suc 𝑚
12 vx . . . . . . . . . . 11 setvar 𝑥
1312cv 1286 . . . . . . . . . 10 class 𝑥
149, 6cfv 4981 . . . . . . . . . . 11 class (𝑔𝑚)
1514, 1cfv 4981 . . . . . . . . . 10 class (𝐹‘(𝑔𝑚))
1613, 15wcel 1436 . . . . . . . . 9 wff 𝑥 ∈ (𝐹‘(𝑔𝑚))
1711, 16wa 102 . . . . . . . 8 wff (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))
18 com 4378 . . . . . . . 8 class ω
1917, 8, 18wrex 2356 . . . . . . 7 wff 𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))
20 c0 3275 . . . . . . . . 9 class
217, 20wceq 1287 . . . . . . . 8 wff dom 𝑔 = ∅
2213, 2wcel 1436 . . . . . . . 8 wff 𝑥𝐼
2321, 22wa 102 . . . . . . 7 wff (dom 𝑔 = ∅ ∧ 𝑥𝐼)
2419, 23wo 662 . . . . . 6 wff (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))
2524, 12cab 2071 . . . . 5 class {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))}
264, 5, 25cmpt 3874 . . . 4 class (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})
2726crecs 6023 . . 3 class recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))}))
2827, 18cres 4413 . 2 class (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐼))})) ↾ ω)
293, 28wceq 1287 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  6111  freceq2  6112  frecex  6113  frecfun  6114  nffrec  6115  frec0g  6116  frecfnom  6120  freccllem  6121  frecfcllem  6123  frecsuclem  6125
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