Definition df-subg 13820

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13839), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13834), contains the neutral element of the group (see subg0 13830) and contains the inverses for all of its elements (see subginvcl 13833). (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 13817	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13646	. . 3 class Grp
4	2	cv 1397	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1397	. . . . . 6 class s
7		cress 13146	. . . . . 6 class ↾s
8	4, 6, 7	co 6028	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2202	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13145	. . . . . 6 class Base
11	4, 10	cfv 5333	. . . . 5 class ( Base ` w )
12	11	cpw 3656	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2515	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4155	. 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  13823  subgex  13826