Definition df-subg 15669

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 15687), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 15682), contains the neutral element of the group (see subg0 15678) and contains the inverses for all of its elements (see subginvcl 15681). (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 15666	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 15402	. . 3 class Grp
4	2	cv 1368	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1368	. . . . . 6 class s
7		cress 14167	. . . . . 6 class ↾s
8	4, 6, 7	co 6086	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1756	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 14166	. . . . . 6 class Base
11	4, 10	cfv 5413	. . . . 5 class ( Base ` w )
12	11	cpw 3855	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2714	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4345	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1369	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  15672