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Mirrors > Home > MPE Home > Th. List > df-wrecs | Structured version Visualization version GIF version |
Description: Define the well-ordered recursive function generator. This function takes the usual expressions from recursion theorems and forms a unified definition. Specifically, given a function 𝐹, a relation 𝑅, and a base set 𝐴, this definition generates a function 𝐺 = wrecs(𝑅, 𝐴, 𝐹) that has property that, at any point 𝑥 ∈ 𝐴, (𝐺‘𝑥) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑥))). See wfr1 8175, wfr2 8176, and wfr3 8177. (Contributed by Scott Fenton, 7-Jun-2018.) (Revised by BJ, 27-Oct-2024.) |
Ref | Expression |
---|---|
df-wrecs | ⊢ wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cR | . . 3 class 𝑅 | |
3 | cF | . . 3 class 𝐹 | |
4 | 1, 2, 3 | cwrecs 8136 | . 2 class wrecs(𝑅, 𝐴, 𝐹) |
5 | c2nd 7839 | . . . 4 class 2nd | |
6 | 3, 5 | ccom 5594 | . . 3 class (𝐹 ∘ 2nd ) |
7 | 1, 2, 6 | cfrecs 8105 | . 2 class frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) |
8 | 4, 7 | wceq 1539 | 1 wff wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd )) |
Colors of variables: wff setvar class |
This definition is referenced by: dfwrecsOLD 8138 wrecseq123 8139 nfwrecs 8141 csbwrecsg 8146 wfrrel 8169 wfrdmss 8170 wfrdmcl 8171 wfrfun 8172 wfrresex 8173 wfr2a 8174 wfr1 8175 dfrecs3 8212 |
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