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Definition df-frec 5898
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 5842 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 5902 and frecsuc 5907.

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 4254. 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 5908, this definition and df-irdg 5878 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‘(g𝑚))) (dom g = ∅ x 𝐼))})) ↾ 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 5897 . 2 class frec(𝐹, 𝐼)
4 vg . . . . 5 setvar g
5 cvv 2535 . . . . 5 class V
64cv 1227 . . . . . . . . . . 11 class g
76cdm 4272 . . . . . . . . . 10 class dom g
8 vm . . . . . . . . . . . 12 setvar 𝑚
98cv 1227 . . . . . . . . . . 11 class 𝑚
109csuc 4051 . . . . . . . . . 10 class suc 𝑚
117, 10wceq 1228 . . . . . . . . 9 wff dom g = suc 𝑚
12 vx . . . . . . . . . . 11 setvar x
1312cv 1227 . . . . . . . . . 10 class x
149, 6cfv 4829 . . . . . . . . . . 11 class (g𝑚)
1514, 1cfv 4829 . . . . . . . . . 10 class (𝐹‘(g𝑚))
1613, 15wcel 1374 . . . . . . . . 9 wff x (𝐹‘(g𝑚))
1711, 16wa 97 . . . . . . . 8 wff (dom g = suc 𝑚 x (𝐹‘(g𝑚)))
18 com 4240 . . . . . . . 8 class 𝜔
1917, 8, 18wrex 2285 . . . . . . 7 wff 𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚)))
20 c0 3201 . . . . . . . . 9 class
217, 20wceq 1228 . . . . . . . 8 wff dom g = ∅
2213, 2wcel 1374 . . . . . . . 8 wff x 𝐼
2321, 22wa 97 . . . . . . 7 wff (dom g = ∅ x 𝐼)
2419, 23wo 616 . . . . . 6 wff (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x 𝐼))
2524, 12cab 2008 . . . . 5 class {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x 𝐼))}
264, 5, 25cmpt 3792 . . . 4 class (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x 𝐼))})
2726crecs 5841 . . 3 class recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x 𝐼))}))
2827, 18cres 4274 . 2 class (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x 𝐼))})) ↾ 𝜔)
293, 28wceq 1228 1 wff frec(𝐹, 𝐼) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x 𝐼))})) ↾ 𝜔)
Colors of variables: wff set class
This definition is referenced by:  frecfnom  5901  frec0g  5902  frecsuclem1  5903  frecsuclem2  5905
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