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Definition df-oexp 7849
 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 7842 . 2 class o
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 5976 . . 3 class On
52cv 1600 . . . . 5 class 𝑥
6 c0 4140 . . . . 5 class
75, 6wceq 1601 . . . 4 wff 𝑥 = ∅
8 c1o 7836 . . . . 5 class 1o
93cv 1600 . . . . 5 class 𝑦
108, 9cdif 3788 . . . 4 class (1o𝑦)
11 vz . . . . . . 7 setvar 𝑧
12 cvv 3397 . . . . . . 7 class V
1311cv 1600 . . . . . . . 8 class 𝑧
14 comu 7841 . . . . . . . 8 class ·o
1513, 5, 14co 6922 . . . . . . 7 class (𝑧 ·o 𝑥)
1611, 12, 15cmpt 4965 . . . . . 6 class (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
1716, 8crdg 7788 . . . . 5 class rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)
189, 17cfv 6135 . . . 4 class (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)
197, 10, 18cif 4306 . . 3 class if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
202, 3, 4, 4, 19cmpt2 6924 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1o𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)))
211, 20wceq 1601 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  7874  oev  7878
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