Definition df-subg 13923

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13942), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13937), contains the neutral element of the group (see subg0 13933) and contains the inverses for all of its elements (see subginvcl 13936). (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 13920	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13755	. . 3 class Grp
4	2	cv 1397	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1397	. . . . . 6 class s
7		cress 13297	. . . . . 6 class ↾s
8	4, 6, 7	co 6058	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2205	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13296	. . . . . 6 class Base
11	4, 10	cfv 5357	. . . . 5 class ( Base ` w )
12	11	cpw 3674	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2526	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4176	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1398	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13926  subgex  13929