Definition df-recs 6284

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 6349 for more details on why this definition is desirable. Unlike df-irdg 6349 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 6314 and tfri2d 6315 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 6283	. 2 class recs(𝐹)
3		vf	. . . . . . . 8 setvar 𝑓
4	3	cv 1347	. . . . . . 7 class 𝑓
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1347	. . . . . . 7 class 𝑥
7	4, 6	wfn 5193	. . . . . 6 wff 𝑓 Fn 𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1347	. . . . . . . . 9 class 𝑦
10	9, 4	cfv 5198	. . . . . . . 8 class (𝑓‘𝑦)
11	4, 9	cres 4613	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
12	11, 1	cfv 5198	. . . . . . . 8 class (𝐹‘(𝑓 ↾ 𝑦))
13	10, 12	wceq 1348	. . . . . . 7 wff (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))
14	13, 8, 6	wral 2448	. . . . . 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 4348	. . . . 5 class On
17	15, 5, 16	wrex 2449	. . . 4 wff ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))
18	17, 3	cab 2156	. . 3 class {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
19	18	cuni 3796	. 2 class ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
20	2, 19	wceq 1348	1 wff recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

This definition is referenced by:  recseq  6285  nfrecs  6286  recsfval  6294  tfrlem9  6298  tfr0dm  6301  tfr1onlemssrecs  6318  tfrcllemssrecs  6331