Definition df-oexpi 6425

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 e. 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 e. On , y e. On |-> ( rec ( ( z e. _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 6418	. 2 class ↑o
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		con0 4365	. . 3 class On
5	3	cv 1352	. . . 4 class y
6		vz	. . . . . 6 setvar z
7		cvv 2739	. . . . . 6 class _V
8	6	cv 1352	. . . . . . 7 class z
9	2	cv 1352	. . . . . . 7 class x
10		comu 6417	. . . . . . 7 class .o
11	8, 9, 10	co 5877	. . . . . 6 class ( z .o x )
12	6, 7, 11	cmpt 4066	. . . . 5 class ( z e. _V |-> ( z .o x ) )
13		c1o 6412	. . . . 5 class 1o
14	12, 13	crdg 6372	. . . 4 class rec ( ( z e. _V |-> ( z .o x ) ) , 1o )
15	5, 14	cfv 5218	. . 3 class ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y )
16	2, 3, 4, 4, 15	cmpo 5879	. 2 class ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )
17	1, 16	wceq 1353	1 wff ↑o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )

This definition is referenced by:  fnoei  6455  oeiexg  6456  oeiv  6459