Definition df-recs 6398

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 6463 for more details on why this definition is desirable. Unlike df-irdg 6463 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 6428 and tfri2d 6429 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 6397	. 2 class recs(𝐹)
3		vf	. . . . . . . 8 setvar 𝑓
4	3	cv 1372	. . . . . . 7 class 𝑓
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1372	. . . . . . 7 class 𝑥
7	4, 6	wfn 5271	. . . . . 6 wff 𝑓 Fn 𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1372	. . . . . . . . 9 class 𝑦
10	9, 4	cfv 5276	. . . . . . . 8 class (𝑓‘𝑦)
11	4, 9	cres 4681	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
12	11, 1	cfv 5276	. . . . . . . 8 class (𝐹‘(𝑓 ↾ 𝑦))
13	10, 12	wceq 1373	. . . . . . 7 wff (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))
14	13, 8, 6	wral 2485	. . . . . 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 4414	. . . . 5 class On
17	15, 5, 16	wrex 2486	. . . 4 wff ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
18	17, 3	cab 2192	. . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
19	18	cuni 3852	. 2 class ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
20	2, 19	wceq 1373	1 wff recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

This definition is referenced by:  recseq  6399  nfrecs  6400  recsfval  6408  tfrlem9  6412  tfr0dm  6415  tfr1onlemssrecs  6432  tfrcllemssrecs  6445