Definition df-subg 14026

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 14045), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14040), contains the neutral element of the group (see subg0 14036) and contains the inverses for all of its elements (see subginvcl 14039). (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 14023	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13858	. . 3 class Grp
4	2	cv 1401	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1401	. . . . . 6 class s
7		cress 13405	. . . . . 6 class ↾s
8	4, 6, 7	co 6085	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2209	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13404	. . . . . 6 class Base
11	4, 10	cfv 5377	. . . . 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 4192	. 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  14029  subgex  14032