Definition df-subg 16807

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16825), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16820), contains the neutral element of the group (see subg0 16816) and contains the inverses for all of its elements (see subginvcl 16819). (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 16804	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 16662	. . 3 class Grp
4	2	cv 1437	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1437	. . . . . 6 class s
7		cress 15115	. . . . . 6 class ↾s
8	4, 6, 7	co 6303	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1869	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 15114	. . . . . 6 class Base
11	4, 10	cfv 5599	. . . . 5 class ( Base ` w )
12	11	cpw 3980	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2780	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4480	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1438	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  16810