Definition df-subg 16814

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16832), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16827), contains the neutral element of the group (see subg0 16823) and contains the inverses for all of its elements (see subginvcl 16826). (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 16811	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 16669	. . 3 class Grp
4	2	cv 1443	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1443	. . . . . 6 class s
7		cress 15122	. . . . . 6 class ↾s
8	4, 6, 7	co 6290	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1887	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 15121	. . . . . 6 class Base
11	4, 10	cfv 5582	. . . . 5 class ( Base ` w )
12	11	cpw 3951	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2741	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4461	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1444	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  16817