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Definition df-relexp 13980
Description: Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 22-May-2020.)
Assertion
Ref Expression
df-relexp 𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
Distinct variable group:   𝑛,𝑟,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-relexp
StepHypRef Expression
1 crelexp 13979 . 2 class 𝑟
2 vr . . 3 setvar 𝑟
3 vn . . 3 setvar 𝑛
4 cvv 3340 . . 3 class V
5 cn0 11504 . . 3 class 0
63cv 1631 . . . . 5 class 𝑛
7 cc0 10148 . . . . 5 class 0
86, 7wceq 1632 . . . 4 wff 𝑛 = 0
9 cid 5173 . . . . 5 class I
102cv 1631 . . . . . . 7 class 𝑟
1110cdm 5266 . . . . . 6 class dom 𝑟
1210crn 5267 . . . . . 6 class ran 𝑟
1311, 12cun 3713 . . . . 5 class (dom 𝑟 ∪ ran 𝑟)
149, 13cres 5268 . . . 4 class ( I ↾ (dom 𝑟 ∪ ran 𝑟))
15 vx . . . . . . 7 setvar 𝑥
16 vy . . . . . . 7 setvar 𝑦
1715cv 1631 . . . . . . . 8 class 𝑥
1817, 10ccom 5270 . . . . . . 7 class (𝑥𝑟)
1915, 16, 4, 4, 18cmpt2 6816 . . . . . 6 class (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟))
20 vz . . . . . . 7 setvar 𝑧
2120, 4, 10cmpt 4881 . . . . . 6 class (𝑧 ∈ V ↦ 𝑟)
22 c1 10149 . . . . . 6 class 1
2319, 21, 22cseq 13015 . . . . 5 class seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))
246, 23cfv 6049 . . . 4 class (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)
258, 14, 24cif 4230 . . 3 class if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))
262, 3, 4, 5, 25cmpt2 6816 . 2 class (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
271, 26wceq 1632 1 wff 𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
Colors of variables: wff setvar class
This definition is referenced by:  relexp0g  13981  relexpsucnnr  13984  relexp1g  13985
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