Definition df-subg 16522

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16540), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16535), contains the neutral element of the group (see subg0 16531) and contains the inverses for all of its elements (see subginvcl 16534). (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 16519	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 16377	. . 3 class Grp
4	2	cv 1404	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1404	. . . . . 6 class s
7		cress 14842	. . . . . 6 class ↾s
8	4, 6, 7	co 6278	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1842	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 14841	. . . . . 6 class Base
11	4, 10	cfv 5569	. . . . 5 class ( Base ` w )
12	11	cpw 3955	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2758	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4453	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1405	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  16525