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Definition df-oexpi 6038
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 6031 . 2 class 𝑜
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 con0 4128 . . 3 class On
53cv 1258 . . . 4 class 𝑦
6 vz . . . . . 6 setvar 𝑧
7 cvv 2574 . . . . . 6 class V
86cv 1258 . . . . . . 7 class 𝑧
92cv 1258 . . . . . . 7 class 𝑥
10 comu 6030 . . . . . . 7 class ·𝑜
118, 9, 10co 5540 . . . . . 6 class (𝑧 ·𝑜 𝑥)
126, 7, 11cmpt 3846 . . . . 5 class (𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥))
13 c1o 6025 . . . . 5 class 1𝑜
1412, 13crdg 5987 . . . 4 class rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)
155, 14cfv 4930 . . 3 class (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦)
162, 3, 4, 4, 15cmpt2 5542 . 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜 𝑥)), 1𝑜)‘𝑦))
171, 16wceq 1259 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  6063  oeiexg  6064  oeiv  6067
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