Definition df-subg 14022

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 14041), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14036), contains the neutral element of the group (see subg0 14032) and contains the inverses for all of its elements (see subginvcl 14035). (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 14019	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13854	. . 3 class Grp
4	2	cv 1401	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1401	. . . . . 6 class s
7		cress 13402	. . . . . 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 13401	. . . . . 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  14025  subgex  14028