Definition df-subg 13243

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13262), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13257), contains the neutral element of the group (see subg0 13253) and contains the inverses for all of its elements (see subginvcl 13256). (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 13240	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13075	. . 3 class Grp
4	2	cv 1363	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1363	. . . . . 6 class s
7		cress 12622	. . . . . 6 class ↾s
8	4, 6, 7	co 5919	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2164	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 12621	. . . . . 6 class Base
11	4, 10	cfv 5255	. . . . 5 class ( Base ` w )
12	11	cpw 3602	. . . 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 4091	. 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  13246  subgex  13249