Definition df-subg 13424

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13443), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13438), contains the neutral element of the group (see subg0 13434) and contains the inverses for all of its elements (see subginvcl 13437). (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 13421	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13250	. . 3 class Grp
4	2	cv 1371	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1371	. . . . . 6 class s
7		cress 12752	. . . . . 6 class ↾s
8	4, 6, 7	co 5934	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2175	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 12751	. . . . . 6 class Base
11	4, 10	cfv 5268	. . . . 5 class ( Base ` w )
12	11	cpw 3615	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2487	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4104	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1372	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13427  subgex  13430