Definition df-oexpi 6508

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 6501	. 2 class ↑o
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		con0 4410	. . 3 class On
5	3	cv 1372	. . . 4 class y
6		vz	. . . . . 6 setvar z
7		cvv 2772	. . . . . 6 class _V
8	6	cv 1372	. . . . . . 7 class z
9	2	cv 1372	. . . . . . 7 class x
10		comu 6500	. . . . . . 7 class .o
11	8, 9, 10	co 5944	. . . . . 6 class ( z .o x )
12	6, 7, 11	cmpt 4105	. . . . 5 class ( z e. _V |-> ( z .o x ) )
13		c1o 6495	. . . . 5 class 1o
14	12, 13	crdg 6455	. . . 4 class rec ( ( z e. _V |-> ( z .o x ) ) , 1o )
15	5, 14	cfv 5271	. . 3 class ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y )
16	2, 3, 4, 4, 15	cmpo 5946	. 2 class ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )
17	1, 16	wceq 1373	1 wff ↑o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )

This definition is referenced by:  fnoei  6538  oeiexg  6539  oeiv  6542