Definition df-recs 6466

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 6531 for more details on why this definition is desirable. Unlike df-irdg 6531 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 6496 and tfri2d 6497 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 6465	. 2 class recs(𝐹)
3		vf	. . . . . . . 8 setvar 𝑓
4	3	cv 1394	. . . . . . 7 class 𝑓
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1394	. . . . . . 7 class 𝑥
7	4, 6	wfn 5319	. . . . . 6 wff 𝑓 Fn 𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1394	. . . . . . . . 9 class 𝑦
10	9, 4	cfv 5324	. . . . . . . 8 class (𝑓‘𝑦)
11	4, 9	cres 4725	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
12	11, 1	cfv 5324	. . . . . . . 8 class (𝐹‘(𝑓 ↾ 𝑦))
13	10, 12	wceq 1395	. . . . . . 7 wff (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))
14	13, 8, 6	wral 2508	. . . . . 6 wff ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))
15	7, 14	wa 104	. . . . 5 wff (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
16		con0 4458	. . . . 5 class On
17	15, 5, 16	wrex 2509	. . . 4 wff ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
18	17, 3	cab 2215	. . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
19	18	cuni 3891	. 2 class ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
20	2, 19	wceq 1395	1 wff recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

This definition is referenced by:  recseq  6467  nfrecs  6468  recsfval  6476  tfrlem9  6480  tfr0dm  6483  tfr1onlemssrecs  6500  tfrcllemssrecs  6513