Definition df-subg 14032

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 14051), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14046), contains the neutral element of the group (see subg0 14042) and contains the inverses for all of its elements (see subginvcl 14045). (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 14029	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13864	. . 3 class Grp
4	2	cv 1401	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1401	. . . . . 6 class s
7		cress 13411	. . . . . 6 class ↾s
8	4, 6, 7	co 6088	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2209	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13410	. . . . . 6 class Base
11	4, 10	cfv 5380	. . . . 5 class ( Base ` w )
12	11	cpw 3688	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2532	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4193	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1402	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is used by:  issubg  14035  subgex  14038