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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 6013 for more details on why
this definition is desirable. Unlike df-irdg 6013 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 5978 and
tfri2d 5979 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 5947 | . 2 class recs(𝐹) |
3 | vf | . . . . . . . 8 setvar 𝑓 | |
4 | 3 | cv 1284 | . . . . . . 7 class 𝑓 |
5 | vx | . . . . . . . 8 setvar 𝑥 | |
6 | 5 | cv 1284 | . . . . . . 7 class 𝑥 |
7 | 4, 6 | wfn 4921 | . . . . . 6 wff 𝑓 Fn 𝑥 |
8 | vy | . . . . . . . . . 10 setvar 𝑦 | |
9 | 8 | cv 1284 | . . . . . . . . 9 class 𝑦 |
10 | 9, 4 | cfv 4926 | . . . . . . . 8 class (𝑓‘𝑦) |
11 | 4, 9 | cres 4367 | . . . . . . . . 9 class (𝑓 ↾ 𝑦) |
12 | 11, 1 | cfv 4926 | . . . . . . . 8 class (𝐹‘(𝑓 ↾ 𝑦)) |
13 | 10, 12 | wceq 1285 | . . . . . . 7 wff (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) |
14 | 13, 8, 6 | wral 2349 | . . . . . 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 4120 | . . . . 5 class On | |
17 | 15, 5, 16 | wrex 2350 | . . . 4 wff ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) |
18 | 17, 3 | cab 2068 | . . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
19 | 18 | cuni 3603 | . 2 class ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
20 | 2, 19 | wceq 1285 | 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 5949 nfrecs 5950 recsfval 5958 tfrlem9 5962 tfr0dm 5965 tfr1onlemssrecs 5982 tfrcllemssrecs 5995 |
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