Definition df-recs 5861

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 5897 for more details on why this definition is desirable. Unlike df-irdg 5897 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 5890 and tfri2d 5891 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 5860	. 2 class recs(𝐹)
3		vf	. . . . . . . 8 setvar f
4	3	cv 1241	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1241	. . . . . . 7 class x
7	4, 6	wfn 4840	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1241	. . . . . . . . 9 class y
10	9, 4	cfv 4845	. . . . . . . 8 class (f‘y)
11	4, 9	cres 4290	. . . . . . . . 9 class (f ↾ y)
12	11, 1	cfv 4845	. . . . . . . 8 class (𝐹‘(f ↾ y))
13	10, 12	wceq 1242	. . . . . . 7 wff (f‘y) = (𝐹‘(f ↾ y))
14	13, 8, 6	wral 2300	. . . . . 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 4066	. . . . 5 class On
17	15, 5, 16	wrex 2301	. . . 4 wff ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))
18	17, 3	cab 2023	. . 3 class {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
19	18	cuni 3571	. 2 class ∪ {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
20	2, 19	wceq 1242	1 wff recs(𝐹) = ∪ {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}

This definition is referenced by:  recseq  5862  nfrecs  5863  recsfval  5872  tfrlem9  5876  tfr0  5878