Definition df-subg 16892

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16910), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16905), contains the neutral element of the group (see subg0 16901) and contains the inverses for all of its elements (see subginvcl 16904). (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 16889	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 16747	. . 3 class Grp
4	2	cv 1451	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1451	. . . . . 6 class s
7		cress 15200	. . . . . 6 class ↾s
8	4, 6, 7	co 6308	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1904	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 15199	. . . . . 6 class Base
11	4, 10	cfv 5589	. . . . 5 class ( Base ` w )
12	11	cpw 3942	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2760	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4454	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1452	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  16895