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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 5897 for more details on why
this definition is desirable. Unlike df-irdg 5897 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 5890 and
tfri2d 5891 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 5860 | . 2 class recs(𝐹) |
3 | vf | . . . . . . . 8 setvar f | |
4 | 3 | cv 1241 | . . . . . . 7 class f |
5 | vx | . . . . . . . 8 setvar x | |
6 | 5 | cv 1241 | . . . . . . 7 class x |
7 | 4, 6 | wfn 4840 | . . . . . 6 wff f Fn x |
8 | vy | . . . . . . . . . 10 setvar y | |
9 | 8 | cv 1241 | . . . . . . . . 9 class y |
10 | 9, 4 | cfv 4845 | . . . . . . . 8 class (f‘y) |
11 | 4, 9 | cres 4290 | . . . . . . . . 9 class (f ↾ y) |
12 | 11, 1 | cfv 4845 | . . . . . . . 8 class (𝐹‘(f ↾ y)) |
13 | 10, 12 | wceq 1242 | . . . . . . 7 wff (f‘y) = (𝐹‘(f ↾ y)) |
14 | 13, 8, 6 | wral 2300 | . . . . . 6 wff ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)) |
15 | 7, 14 | wa 97 | . . . . 5 wff (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y))) |
16 | con0 4066 | . . . . 5 class On | |
17 | 15, 5, 16 | wrex 2301 | . . . 4 wff ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y))) |
18 | 17, 3 | cab 2023 | . . 3 class {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))} |
19 | 18 | cuni 3571 | . 2 class ∪ {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))} |
20 | 2, 19 | wceq 1242 | 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 5862 nfrecs 5863 recsfval 5872 tfrlem9 5876 tfr0 5878 |
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