Definition df-wrecs

Description: Here we define the well-founded recursive function generator. This function takes the usual expressions from recursion theorems and forms a unified definition. Specifically, given a function F , a relationship R , and a base set A , this definition generates a function G = wrecs ( R , A , F ) that has property that, at any point x e. A , ( Gx ) = ( F` ( G |`Pred ( R , A , x ) ) ) . See wfr1 , wfr2 , and wfr3 . (Contributed by Scott Fenton, 7-Jun-2018) (New usage is discouraged.)

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

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cR	⊢ 𝑅
1		cA	⊢ 𝐴
2		cF	⊢ 𝐹
3	1 0 2	cwrecs	⊢ wrecs ( 𝑅 , 𝐴 , 𝐹 )
4		vf	⊢ 𝑓
5		vx	⊢ 𝑥
6	4	cv	⊢ 𝑓
7	5	cv	⊢ 𝑥
8	6 7	wfn	⊢ 𝑓 Fn 𝑥
9	7 1	wss	⊢ 𝑥 ⊆ 𝐴
10		vy	⊢ 𝑦
11	10	cv	⊢ 𝑦
12	1 0 11	cpred	⊢ Pred ( 𝑅 , 𝐴 , 𝑦 )
13	12 7	wss	⊢ Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥
14	13 10 7	wral	⊢ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥
15	9 14	wa	⊢ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 )
16	11 6	cfv	⊢ ( 𝑓 ‘ 𝑦 )
17	6 12	cres	⊢ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) )
18	17 2	cfv	⊢ ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) )
19	16 18	wceq	⊢ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) )
20	19 10 7	wral	⊢ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) )
21	8 15 20	w3a	⊢ ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
22	21 5	wex	⊢ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) )
23	22 4	cab	⊢ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
24	23	cuni	⊢ ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
25	3 24	wceq	⊢ wrecs ( 𝑅 , 𝐴 , 𝐹 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }