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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 6275 for more details on why
this definition is desirable. Unlike df-irdg 6275 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 6240 and
tfri2d 6241 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 6209 | . 2 class recs(𝐹) |
3 | vf | . . . . . . . 8 setvar 𝑓 | |
4 | 3 | cv 1331 | . . . . . . 7 class 𝑓 |
5 | vx | . . . . . . . 8 setvar 𝑥 | |
6 | 5 | cv 1331 | . . . . . . 7 class 𝑥 |
7 | 4, 6 | wfn 5126 | . . . . . 6 wff 𝑓 Fn 𝑥 |
8 | vy | . . . . . . . . . 10 setvar 𝑦 | |
9 | 8 | cv 1331 | . . . . . . . . 9 class 𝑦 |
10 | 9, 4 | cfv 5131 | . . . . . . . 8 class (𝑓‘𝑦) |
11 | 4, 9 | cres 4549 | . . . . . . . . 9 class (𝑓 ↾ 𝑦) |
12 | 11, 1 | cfv 5131 | . . . . . . . 8 class (𝐹‘(𝑓 ↾ 𝑦)) |
13 | 10, 12 | wceq 1332 | . . . . . . 7 wff (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) |
14 | 13, 8, 6 | wral 2417 | . . . . . 6 wff ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) |
15 | 7, 14 | wa 103 | . . . . 5 wff (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) |
16 | con0 4293 | . . . . 5 class On | |
17 | 15, 5, 16 | wrex 2418 | . . . 4 wff ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) |
18 | 17, 3 | cab 2126 | . . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
19 | 18 | cuni 3744 | . 2 class ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
20 | 2, 19 | wceq 1332 | 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 6211 nfrecs 6212 recsfval 6220 tfrlem9 6224 tfr0dm 6227 tfr1onlemssrecs 6244 tfrcllemssrecs 6257 |
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