Definition df-oexpi 6587

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.

We do not yet have an extensive development of ordinal exponentiation. For background on ordinal exponentiation without excluded middle, see Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu (2025), "Ordinal Exponentiation in Homotopy Type Theory", arXiv:2501.14542 , https://arxiv.org/abs/2501.14542 which is formalized in the TypeTopology proof library at https://ordinal-exponentiation-hott.github.io/.

(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 6580	. 2 class ↑o
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		con0 4460	. . 3 class On
5	3	cv 1396	. . . 4 class 𝑦
6		vz	. . . . . 6 setvar 𝑧
7		cvv 2802	. . . . . 6 class V
8	6	cv 1396	. . . . . . 7 class 𝑧
9	2	cv 1396	. . . . . . 7 class 𝑥
10		comu 6579	. . . . . . 7 class ·o
11	8, 9, 10	co 6017	. . . . . 6 class (𝑧 ·o 𝑥)
12	6, 7, 11	cmpt 4150	. . . . 5 class (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
13		c1o 6574	. . . . 5 class 1o
14	12, 13	crdg 6534	. . . 4 class rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)
15	5, 14	cfv 5326	. . 3 class (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)
16	2, 3, 4, 4, 15	cmpo 6019	. 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
17	1, 16	wceq 1397	1 wff ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))

This definition is referenced by:  fnoei  6619  oeiexg  6620  oeiv  6623