Definition df-seqom 8373

Description: Index-aware recursive definitions over ω. A mashup of df-rdg 8335 and df-seq 13911, this allows for recursive definitions that use an index in the recursion in cases where Infinity is not admitted. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Assertion

Ref	Expression
df-seqom	⊢ seqω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)

Distinct variable groups:   𝑖,𝐹,𝑣   𝑖,𝐼,𝑣

Detailed syntax breakdown of Definition df-seqom

Step	Hyp	Ref	Expression
1		cF	. . 3 class 𝐹
2		cI	. . 3 class 𝐼
3	1, 2	cseqom 8372	. 2 class seqω(𝐹, 𝐼)
4		vi	. . . . 5 setvar 𝑖
5		vv	. . . . 5 setvar 𝑣
6		com 7802	. . . . 5 class ω
7		cvv 3437	. . . . 5 class V
8	4	cv 1540	. . . . . . 7 class 𝑖
9	8	csuc 6313	. . . . . 6 class suc 𝑖
10	5	cv 1540	. . . . . . 7 class 𝑣
11	8, 10, 1	co 7352	. . . . . 6 class (𝑖𝐹𝑣)
12	9, 11	cop 4581	. . . . 5 class ⟨suc 𝑖, (𝑖𝐹𝑣)⟩
13	4, 5, 6, 7, 12	cmpo 7354	. . . 4 class (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
14		c0 4282	. . . . 5 class ∅
15		cid 5513	. . . . . 6 class I
16	2, 15	cfv 6486	. . . . 5 class ( I ‘𝐼)
17	14, 16	cop 4581	. . . 4 class ⟨∅, ( I ‘𝐼)⟩
18	13, 17	crdg 8334	. . 3 class rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
19	18, 6	cima 5622	. 2 class (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
20	3, 19	wceq 1541	1 wff seqω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)

This definition is referenced by:  seqomeq12  8379  fnseqom  8380  seqom0g  8381  seqomsuc  8382