Definition df-recs 6308

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 6373 for more details on why this definition is desirable. Unlike df-irdg 6373 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 6338 and tfri2d 6339 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 6307	. 2 class recs(𝐹)
3		vf	. . . . . . . 8 setvar 𝑓
4	3	cv 1352	. . . . . . 7 class 𝑓
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1352	. . . . . . 7 class 𝑥
7	4, 6	wfn 5213	. . . . . 6 wff 𝑓 Fn 𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1352	. . . . . . . . 9 class 𝑦
10	9, 4	cfv 5218	. . . . . . . 8 class (𝑓‘𝑦)
11	4, 9	cres 4630	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
12	11, 1	cfv 5218	. . . . . . . 8 class (𝐹‘(𝑓 ↾ 𝑦))
13	10, 12	wceq 1353	. . . . . . 7 wff (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))
14	13, 8, 6	wral 2455	. . . . . 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 4365	. . . . 5 class On
17	15, 5, 16	wrex 2456	. . . 4 wff ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
18	17, 3	cab 2163	. . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
19	18	cuni 3811	. 2 class ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
20	2, 19	wceq 1353	1 wff recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

This definition is referenced by:  recseq  6309  nfrecs  6310  recsfval  6318  tfrlem9  6322  tfr0dm  6325  tfr1onlemssrecs  6342  tfrcllemssrecs  6355