Definition df-subg 13581

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13600), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13595), contains the neutral element of the group (see subg0 13591) and contains the inverses for all of its elements (see subginvcl 13594). (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 13578	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13407	. . 3 class Grp
4	2	cv 1372	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1372	. . . . . 6 class s
7		cress 12908	. . . . . 6 class ↾s
8	4, 6, 7	co 5957	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2177	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 12907	. . . . . 6 class Base
11	4, 10	cfv 5280	. . . . 5 class ( Base ` w )
12	11	cpw 3621	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2489	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4113	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1373	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13584  subgex  13587