Definition df-oexpi 6390

Description: Define the ordinal exponentiation operation.

This definition is similar to a conventional definition of exponentiation except that it defines ∅ ↑_o 𝐴 to be 1_o 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

Step	Hyp	Ref	Expression
1		coei 6383	. 2 class ↑o
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		con0 4341	. . 3 class On
5	3	cv 1342	. . . 4 class 𝑦
6		vz	. . . . . 6 setvar 𝑧
7		cvv 2726	. . . . . 6 class V
8	6	cv 1342	. . . . . . 7 class 𝑧
9	2	cv 1342	. . . . . . 7 class 𝑥
10		comu 6382	. . . . . . 7 class ·o
11	8, 9, 10	co 5842	. . . . . 6 class (𝑧 ·o 𝑥)
12	6, 7, 11	cmpt 4043	. . . . 5 class (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
13		c1o 6377	. . . . 5 class 1o
14	12, 13	crdg 6337	. . . 4 class rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)
15	5, 14	cfv 5188	. . 3 class (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)
16	2, 3, 4, 4, 15	cmpo 5844	. 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
17	1, 16	wceq 1343	1 wff ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))

This definition is referenced by:  fnoei  6420  oeiexg  6421  oeiv  6424