Definition df-subg 16000

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16018), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16013), contains the neutral element of the group (see subg0 16009) and contains the inverses for all of its elements (see subginvcl 16012). (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 15997	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 15726	. . 3 class Grp
4	2	cv 1378	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1378	. . . . . 6 class s
7		cress 14490	. . . . . 6 class ↾s
8	4, 6, 7	co 6283	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1767	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 14489	. . . . . 6 class Base
11	4, 10	cfv 5587	. . . . 5 class ( Base ` w )
12	11	cpw 4010	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2818	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4505	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1379	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  16003