Definition df-ginv 27349

Description: Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
df-ginv	|- inv = ( g e. GrpOp |-> ( x e. ran g |-> ( iota_ z e. ran g ( z g x ) = (GId ` g ) ) ) )

Distinct variable group:   x, g, z

Detailed syntax breakdown of Definition df-ginv

Step	Hyp	Ref	Expression
1		cgn 27345	. 2 class inv
2		vg	. . 3 setvar g
3		cgr 27343	. . 3 class GrpOp
4		vx	. . . 4 setvar x
5	2	cv 1482	. . . . 5 class g
6	5	crn 5115	. . . 4 class ran g
7		vz	. . . . . . . 8 setvar z
8	7	cv 1482	. . . . . . 7 class z
9	4	cv 1482	. . . . . . 7 class x
10	8, 9, 5	co 6650	. . . . . 6 class ( z g x )
11		cgi 27344	. . . . . . 7 class GId
12	5, 11	cfv 5888	. . . . . 6 class (GId ` g )
13	10, 12	wceq 1483	. . . . 5 wff ( z g x ) = (GId ` g )
14	13, 7, 6	crio 6610	. . . 4 class ( iota_ z e. ran g ( z g x ) = (GId ` g ) )
15	4, 6, 14	cmpt 4729	. . 3 class ( x e. ran g |-> ( iota_ z e. ran g ( z g x ) = (GId ` g ) ) )
16	2, 3, 15	cmpt 4729	. 2 class ( g e. GrpOp |-> ( x e. ran g |-> ( iota_ z e. ran g ( z g x ) = (GId ` g ) ) ) )
17	1, 16	wceq 1483	1 wff inv = ( g e. GrpOp |-> ( x e. ran g |-> ( iota_ z e. ran g ( z g x ) = (GId ` g ) ) ) )

This definition is referenced by:  grpoinvfval  27376