Definition df-recs 6210

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.)

Assertion

Ref	Expression
df-recs	⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Distinct variable group:   𝑓,𝐹,𝑥,𝑦

Detailed syntax breakdown of Definition df-recs

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)))}

This definition is referenced by:  recseq  6211  nfrecs  6212  recsfval  6220  tfrlem9  6224  tfr0dm  6227  tfr1onlemssrecs  6244  tfrcllemssrecs  6257