Definition df-recs 5830

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 5829	. 2 class recs(𝐹)
3		vf	. . . . . . . 8 setvar f
4	3	cv 1222	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1222	. . . . . . 7 class x
7	4, 6	wfn 4812	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1222	. . . . . . . . 9 class y
10	9, 4	cfv 4817	. . . . . . . 8 class (f‘y)
11	4, 9	cres 4262	. . . . . . . . 9 class (f ↾ y)
12	11, 1	cfv 4817	. . . . . . . 8 class (𝐹‘(f ↾ y))
13	10, 12	wceq 1223	. . . . . . 7 wff (f‘y) = (𝐹‘(f ↾ y))
14	13, 8, 6	wral 2275	. . . . . 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 4038	. . . . 5 class On
17	15, 5, 16	wrex 2276	. . . 4 wff ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))
18	17, 3	cab 1999	. . 3 class {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
19	18	cuni 3543	. 2 class ∪ {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
20	2, 19	wceq 1223	1 wff recs(𝐹) = ∪ {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}

This definition is referenced by:  recseq  5831  nfrecs  5832  recsfval  5841  tfrlem9  5845