Definition df-subg 13707

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13726), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13721), contains the neutral element of the group (see subg0 13717) and contains the inverses for all of its elements (see subginvcl 13720). (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 13704	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13533	. . 3 class Grp
4	2	cv 1394	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1394	. . . . . 6 class s
7		cress 13033	. . . . . 6 class ↾s
8	4, 6, 7	co 6001	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2200	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13032	. . . . . 6 class Base
11	4, 10	cfv 5318	. . . . 5 class ( Base ` w )
12	11	cpw 3649	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2512	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4145	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1395	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13710  subgex  13713