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Definition df-oexpi 6359
 Description: Define the ordinal exponentiation operation. This definition is similar to a conventional definition of exponentiation except that it defines ∅ ↑o 𝐴 to be 1o for all 𝐴 ∈ On, in order to avoid having different cases for whether the base is ∅ or not. (Contributed by Mario Carneiro, 4-Jul-2019.)
Assertion
Ref Expression
df-oexpi o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-oexpi
StepHypRef Expression
1 coei 6352 . 2 class o
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 4318 . . 3 class On
53cv 1331 . . . 4 class 𝑦
6 vz . . . . . 6 setvar 𝑧
7 cvv 2709 . . . . . 6 class V
86cv 1331 . . . . . . 7 class 𝑧
92cv 1331 . . . . . . 7 class 𝑥
10 comu 6351 . . . . . . 7 class ·o
118, 9, 10co 5814 . . . . . 6 class (𝑧 ·o 𝑥)
126, 7, 11cmpt 4021 . . . . 5 class (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
13 c1o 6346 . . . . 5 class 1o
1412, 13crdg 6306 . . . 4 class rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)
155, 14cfv 5163 . . 3 class (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)
162, 3, 4, 4, 15cmpo 5816 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
171, 16wceq 1332 1 wff o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
 Colors of variables: wff set class This definition is referenced by:  fnoei  6388  oeiexg  6389  oeiv  6392
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