Definition df-oexpi 6693

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 6686	. 2 class ↑o
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		con0 4508	. . 3 class On
5	3	cv 1401	. . . 4 class 𝑦
6		vz	. . . . . 6 setvar 𝑧
7		cvv 2821	. . . . . 6 class V
8	6	cv 1401	. . . . . . 7 class 𝑧
9	2	cv 1401	. . . . . . 7 class 𝑥
10		comu 6685	. . . . . . 7 class ·o
11	8, 9, 10	co 6085	. . . . . 6 class (𝑧 ·o 𝑥)
12	6, 7, 11	cmpt 4192	. . . . 5 class (𝑧 ∈ V ↦ (𝑧 ·o 𝑥))
13		c1o 6680	. . . . 5 class 1o
14	12, 13	crdg 6640	. . . 4 class rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)
15	5, 14	cfv 5377	. . 3 class (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦)
16	2, 3, 4, 4, 15	cmpo 6087	. 2 class (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))
17	1, 16	wceq 1402	1 wff ↑o = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 ·o 𝑥)), 1o)‘𝑦))

This definition is used by:  fnoei  6725  oeiexg  6726  oeiv  6729