Definition df-subg 17591

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17609), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17604), contains the neutral element of the group (see subg0 17600) and contains the inverses for all of its elements (see subginvcl 17603). (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 17588	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 17422	. . 3 class Grp
4	2	cv 1482	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1482	. . . . . 6 class s
7		cress 15858	. . . . . 6 class ↾s
8	4, 6, 7	co 6650	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1990	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 15857	. . . . . 6 class Base
11	4, 10	cfv 5888	. . . . 5 class ( Base ` w )
12	11	cpw 4158	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2916	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4729	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1483	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  17594