Definition df-subg 13240

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13259), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13254), contains the neutral element of the group (see subg0 13250) and contains the inverses for all of its elements (see subginvcl 13253). (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 13237	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13072	. . 3 class Grp
4	2	cv 1363	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1363	. . . . . 6 class s
7		cress 12619	. . . . . 6 class ↾s
8	4, 6, 7	co 5918	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2164	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 12618	. . . . . 6 class Base
11	4, 10	cfv 5254	. . . . 5 class ( Base ` w )
12	11	cpw 3601	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2476	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4090	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1364	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13243  subgex  13246