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Mirrors > Home > ILE Home > Th. List > df-recs | GIF version |
Description: Define a function recs(𝐹)
on On, the class of ordinal
numbers, by transfinite recursion given a rule 𝐹 which sets the next
value given all values so far. See df-irdg 5980 for more details on why
this definition is desirable. Unlike df-irdg 5980 which restricts the
update rule to use only the previous value, this version allows the
update rule to use all previous values, which is why it is
described
as "strong", although it is actually more primitive. See tfri1d 5972 and
tfri2d 5973 for the primary contract of this definition.
(Contributed by Stefan O'Rear, 18-Jan-2015.) |
Ref | Expression |
---|---|
df-recs | ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cF | . . 3 class 𝐹 | |
2 | 1 | crecs 5942 | . 2 class recs(𝐹) |
3 | vf | . . . . . . . 8 setvar 𝑓 | |
4 | 3 | cv 1283 | . . . . . . 7 class 𝑓 |
5 | vx | . . . . . . . 8 setvar 𝑥 | |
6 | 5 | cv 1283 | . . . . . . 7 class 𝑥 |
7 | 4, 6 | wfn 4917 | . . . . . 6 wff 𝑓 Fn 𝑥 |
8 | vy | . . . . . . . . . 10 setvar 𝑦 | |
9 | 8 | cv 1283 | . . . . . . . . 9 class 𝑦 |
10 | 9, 4 | cfv 4922 | . . . . . . . 8 class (𝑓‘𝑦) |
11 | 4, 9 | cres 4365 | . . . . . . . . 9 class (𝑓 ↾ 𝑦) |
12 | 11, 1 | cfv 4922 | . . . . . . . 8 class (𝐹‘(𝑓 ↾ 𝑦)) |
13 | 10, 12 | wceq 1284 | . . . . . . 7 wff (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) |
14 | 13, 8, 6 | wral 2348 | . . . . . 6 wff ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) |
15 | 7, 14 | wa 102 | . . . . 5 wff (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) |
16 | con0 4118 | . . . . 5 class On | |
17 | 15, 5, 16 | wrex 2349 | . . . 4 wff ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) |
18 | 17, 3 | cab 2067 | . . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
19 | 18 | cuni 3601 | . 2 class ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
20 | 2, 19 | wceq 1284 | 1 wff recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
Colors of variables: wff set class |
This definition is referenced by: recseq 5944 nfrecs 5945 recsfval 5954 tfrlem9 5958 tfr0 5960 |
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