Definition df-subg 13300

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13319), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13314), contains the neutral element of the group (see subg0 13310) and contains the inverses for all of its elements (see subginvcl 13313). (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 13297	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13132	. . 3 class Grp
4	2	cv 1363	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1363	. . . . . 6 class s
7		cress 12679	. . . . . 6 class ↾s
8	4, 6, 7	co 5922	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2167	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 12678	. . . . . 6 class Base
11	4, 10	cfv 5258	. . . . 5 class ( Base ` w )
12	11	cpw 3605	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2479	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4094	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1364	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13303  subgex  13306