Definition df-oexp 7566

Description: Define the ordinal exponentiation operation. (Contributed by NM, 30-Dec-2004.)

Assertion

Ref	Expression
df-oexp	|- ^o = ( x e. On , y e. On |-> if ( x = (/) , ( 1o \ y ) , ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) ) )

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-oexp

Step	Hyp	Ref	Expression
1		coe 7559	. 2 class ^o
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		con0 5723	. . 3 class On
5	2	cv 1482	. . . . 5 class x
6		c0 3915	. . . . 5 class (/)
7	5, 6	wceq 1483	. . . 4 wff x = (/)
8		c1o 7553	. . . . 5 class 1o
9	3	cv 1482	. . . . 5 class y
10	8, 9	cdif 3571	. . . 4 class ( 1o \ y )
11		vz	. . . . . . 7 setvar z
12		cvv 3200	. . . . . . 7 class _V
13	11	cv 1482	. . . . . . . 8 class z
14		comu 7558	. . . . . . . 8 class .o
15	13, 5, 14	co 6650	. . . . . . 7 class ( z .o x )
16	11, 12, 15	cmpt 4729	. . . . . 6 class ( z e. _V |-> ( z .o x ) )
17	16, 8	crdg 7505	. . . . 5 class rec ( ( z e. _V |-> ( z .o x ) ) , 1o )
18	9, 17	cfv 5888	. . . 4 class ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y )
19	7, 10, 18	cif 4086	. . 3 class if ( x = (/) , ( 1o \ y ) , ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) )
20	2, 3, 4, 4, 19	cmpt2 6652	. 2 class ( x e. On , y e. On |-> if ( x = (/) , ( 1o \ y ) , ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) ) )
21	1, 20	wceq 1483	1 wff ^o = ( x e. On , y e. On |-> if ( x = (/) , ( 1o \ y ) , ( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) ) )

This definition is referenced by:  fnoe  7590  oev  7594