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Definition df-oexp 8104
Description: Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)
Assertion
Ref Expression
df-oexp o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-oexp
StepHypRef Expression
1 coe 8097 . 2 class o
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 6178 . . 3 class On
52cv 1537 . . . . 5 class 𝑥
6 c0 4276 . . . . 5 class
75, 6wceq 1538 . . . 4 wff 𝑥 = ∅
8 c1o 8091 . . . . 5 class 1o
93cv 1537 . . . . 5 class 𝑦
108, 9cdif 3916 . . . 4 class (1o𝑦)
11 vz . . . . . . 7 setvar 𝑧
12 cvv 3480 . . . . . . 7 class V
1311cv 1537 . . . . . . . 8 class 𝑧
14 comu 8096 . . . . . . . 8 class ·o
1513, 5, 14co 7149 . . . . . . 7 class (𝑧 ·o 𝑥)
1611, 12, 15cmpt 5132 . . . . . 6 class (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
1716, 8crdg 8041 . . . . 5 class rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)
189, 17cfv 6343 . . . 4 class (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)
197, 10, 18cif 4450 . . 3 class if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
202, 3, 4, 4, 19cmpo 7151 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
211, 20wceq 1538 1 wff o = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
Colors of variables: wff setvar class
This definition is referenced by:  fnoe  8131  oev  8135
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