Definition df-subg 16176

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16194), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16189), contains the neutral element of the group (see subg0 16185) and contains the inverses for all of its elements (see subginvcl 16188). (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 16173	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 16031	. . 3 class Grp
4	2	cv 1382	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1382	. . . . . 6 class s
7		cress 14614	. . . . . 6 class ↾s
8	4, 6, 7	co 6281	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1804	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 14613	. . . . . 6 class Base
11	4, 10	cfv 5578	. . . . 5 class ( Base ` w )
12	11	cpw 3997	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2797	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4495	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1383	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  16179