Definition df-oexpi 6312

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. (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 6305	. 2 class ↑o
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		con0 4280	. . 3 class On
5	3	cv 1330	. . . 4 class y
6		vz	. . . . . 6 setvar z
7		cvv 2681	. . . . . 6 class _V
8	6	cv 1330	. . . . . . 7 class z
9	2	cv 1330	. . . . . . 7 class x
10		comu 6304	. . . . . . 7 class .o
11	8, 9, 10	co 5767	. . . . . 6 class ( z .o x )
12	6, 7, 11	cmpt 3984	. . . . 5 class ( z e. _V |-> ( z .o x ) )
13		c1o 6299	. . . . 5 class 1o
14	12, 13	crdg 6259	. . . 4 class rec ( ( z e. _V |-> ( z .o x ) ) , 1o )
15	5, 14	cfv 5118	. . 3 class ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y )
16	2, 3, 4, 4, 15	cmpo 5769	. 2 class ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )
17	1, 16	wceq 1331	1 wff ↑o = ( x e. On , y e. On |-> ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )

This definition is referenced by:  fnoei  6341  oeiexg  6342  oeiv  6345