Definition df-recs 5838

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. If we were assuming the law of the excluded middle, we would then build on top of that a form of recursion which has separate cases for the empty set, successor ordinals, and limit ordinals. This version allows the update rule to use all previous values, which is why it is described as "strong".

(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 5837	. 2 class recs(𝐹)
3		vf	. . . . . . . 8 setvar f
4	3	cv 1225	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1225	. . . . . . 7 class x
7	4, 6	wfn 4820	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1225	. . . . . . . . 9 class y
10	9, 4	cfv 4825	. . . . . . . 8 class (f‘y)
11	4, 9	cres 4270	. . . . . . . . 9 class (f ↾ y)
12	11, 1	cfv 4825	. . . . . . . 8 class (𝐹‘(f ↾ y))
13	10, 12	wceq 1226	. . . . . . 7 wff (f‘y) = (𝐹‘(f ↾ y))
14	13, 8, 6	wral 2280	. . . . . 6 wff ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y))
15	7, 14	wa 97	. . . . 5 wff (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))
16		con0 4045	. . . . 5 class On
17	15, 5, 16	wrex 2281	. . . 4 wff ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))
18	17, 3	cab 2004	. . 3 class {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
19	18	cuni 3550	. 2 class ∪ {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
20	2, 19	wceq 1226	1 wff recs(𝐹) = ∪ {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}

This definition is referenced by:  recseq  5839  nfrecs  5840  recsfval  5849  tfrlem9  5853