Definition df-subg 13035

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13054), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13049), contains the neutral element of the group (see subg0 13045) and contains the inverses for all of its elements (see subginvcl 13048). (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 13032	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 12882	. . 3 class Grp
4	2	cv 1352	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1352	. . . . . 6 class s
7		cress 12465	. . . . . 6 class ↾s
8	4, 6, 7	co 5877	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2148	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 12464	. . . . . 6 class Base
11	4, 10	cfv 5218	. . . . 5 class ( Base ` w )
12	11	cpw 3577	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2459	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4066	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1353	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13038  subgex  13041