Definition df-frecs 32729

Description: This is the definition for the founded recursion generator. Similar to df-wrecs 7805 and df-recs 7867, it is a direct definition form of normally recursive relationships. Unlike the former two definitions, it only requires a founded set-like relationship for its properties, not a well-founded relationship. When this relationship is also a partial ordering, the proof does not use the Axiom of Infinity, but it requires Infinity when the order is not partial. We develop the theorems twice, once with partial ordering and once without. (Contributed by Scott Fenton, 23-Dec-2021.)

Assertion

Ref	Expression
df-frecs	⊢ frecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

Distinct variable groups:   𝑅,𝑓,𝑥,𝑦   𝐴,𝑓,𝑥,𝑦   𝑓,𝐹,𝑥,𝑦

Detailed syntax breakdown of Definition df-frecs

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cR	. . 3 class 𝑅
3		cF	. . 3 class 𝐹
4	1, 2, 3	cfrecs 32728	. 2 class frecs(𝑅, 𝐴, 𝐹)
5		vf	. . . . . . . 8 setvar 𝑓
6	5	cv 1524	. . . . . . 7 class 𝑓
7		vx	. . . . . . . 8 setvar 𝑥
8	7	cv 1524	. . . . . . 7 class 𝑥
9	6, 8	wfn 6227	. . . . . 6 wff 𝑓 Fn 𝑥
10	8, 1	wss 3865	. . . . . . 7 wff 𝑥 ⊆ 𝐴
11		vy	. . . . . . . . . . 11 setvar 𝑦
12	11	cv 1524	. . . . . . . . . 10 class 𝑦
13	1, 2, 12	cpred 6029	. . . . . . . . 9 class Pred(𝑅, 𝐴, 𝑦)
14	13, 8	wss 3865	. . . . . . . 8 wff Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥
15	14, 11, 8	wral 3107	. . . . . . 7 wff ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥
16	10, 15	wa 396	. . . . . 6 wff (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥)
17	12, 6	cfv 6232	. . . . . . . 8 class (𝑓‘𝑦)
18	6, 13	cres 5452	. . . . . . . . 9 class (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))
19	12, 18, 3	co 7023	. . . . . . . 8 class (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
20	17, 19	wceq 1525	. . . . . . 7 wff (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
21	20, 11, 8	wral 3107	. . . . . 6 wff ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
22	9, 16, 21	w3a 1080	. . . . 5 wff (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
23	22, 7	wex 1765	. . . 4 wff ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
24	23, 5	cab 2777	. . 3 class {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
25	24	cuni 4751	. 2 class ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
26	4, 25	wceq 1525	1 wff frecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

This definition is referenced by:  frecseq123  32730  nffrecs  32731  frrlem5  32738