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Definition df-seqom 8071
Description: Index-aware recursive definitions over ω. A mashup of df-rdg 8033 and df-seq 13365, 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
StepHypRef Expression
1 cF . . 3 class 𝐹
2 cI . . 3 class 𝐼
31, 2cseqom 8070 . 2 class seqω(𝐹, 𝐼)
4 vi . . . . 5 setvar 𝑖
5 vv . . . . 5 setvar 𝑣
6 com 7565 . . . . 5 class ω
7 cvv 3469 . . . . 5 class V
84cv 1537 . . . . . . 7 class 𝑖
98csuc 6171 . . . . . 6 class suc 𝑖
105cv 1537 . . . . . . 7 class 𝑣
118, 10, 1co 7140 . . . . . 6 class (𝑖𝐹𝑣)
129, 11cop 4545 . . . . 5 class ⟨suc 𝑖, (𝑖𝐹𝑣)⟩
134, 5, 6, 7, 12cmpo 7142 . . . 4 class (𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩)
14 c0 4265 . . . . 5 class
15 cid 5436 . . . . . 6 class I
162, 15cfv 6334 . . . . 5 class ( I ‘𝐼)
1714, 16cop 4545 . . . 4 class ⟨∅, ( I ‘𝐼)⟩
1813, 17crdg 8032 . . 3 class rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
1918, 6cima 5535 . 2 class (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
203, 19wceq 1538 1 wff seqω(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
Colors of variables: wff setvar class
This definition is referenced by:  seqomeq12  8077  fnseqom  8078  seqom0g  8079  seqomsuc  8080
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