Definition df-subg 13887

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13906), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13901), contains the neutral element of the group (see subg0 13897) and contains the inverses for all of its elements (see subginvcl 13900). (Contributed by Mario Carneiro, 2-Dec-2014.)

Assertion

Ref	Expression
df-subg	|- SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

Distinct variable group:   w, s

Detailed syntax breakdown of Definition df-subg

Step	Hyp	Ref	Expression
1		csubg 13884	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13713	. . 3 class Grp
4	2	cv 1397	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1397	. . . . . 6 class s
7		cress 13213	. . . . . 6 class ↾s
8	4, 6, 7	co 6050	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2203	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13212	. . . . . 6 class Base
11	4, 10	cfv 5352	. . . . 5 class ( Base ` w )
12	11	cpw 3669	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2524	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4171	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1398	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13890  subgex  13893