Definition df-subg 14896

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14914), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14909), contains the neutral element of the group (see subg0 14905) and contains the inverses for all of its elements (see subginvcl 14908). (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 14893	. 2 class SubGrp
2		vw	. . 3 set w
3		cgrp 14640	. . 3 class Grp
4	2	cv 1648	. . . . . 6 class w
5		vs	. . . . . . 7 set s
6	5	cv 1648	. . . . . 6 class s
7		cress 13425	. . . . . 6 class ↾s
8	4, 6, 7	co 6040	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1721	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13424	. . . . . 6 class Base
11	4, 10	cfv 5413	. . . . 5 class ( Base ` w )
12	11	cpw 3759	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2670	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4226	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1649	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  14899