Definition df-subg 13747

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13766), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13761), contains the neutral element of the group (see subg0 13757) and contains the inverses for all of its elements (see subginvcl 13760). (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 13744	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13573	. . 3 class Grp
4	2	cv 1394	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1394	. . . . . 6 class s
7		cress 13073	. . . . . 6 class ↾s
8	4, 6, 7	co 6013	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2200	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13072	. . . . . 6 class Base
11	4, 10	cfv 5324	. . . . 5 class ( Base ` w )
12	11	cpw 3650	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2512	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4148	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1395	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13750  subgex  13753