Definition df-oexpi 6566

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 6559	. 2 class ↑o
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		con0 4453	. . 3 class On
5	3	cv 1394	. . . 4 class 𝑦
6		vz	. . . . . 6 setvar 𝑧
7		cvv 2799	. . . . . 6 class V
8	6	cv 1394	. . . . . . 7 class 𝑧
9	2	cv 1394	. . . . . . 7 class 𝑥
10		comu 6558	. . . . . . 7 class ·o
11	8, 9, 10	co 6000	. . . . . 6 class (𝑧 ·o 𝑥)
12	6, 7, 11	cmpt 4144	. . . . 5 class (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
13		c1o 6553	. . . . 5 class 1o
14	12, 13	crdg 6513	. . . 4 class rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)
15	5, 14	cfv 5317	. . 3 class (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)
16	2, 3, 4, 4, 15	cmpo 6002	. 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
17	1, 16	wceq 1395	1 wff ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))

This definition is referenced by:  fnoei  6596  oeiexg  6597  oeiv  6600