Definition df-subg 13074

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13093), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13088), contains the neutral element of the group (see subg0 13084) and contains the inverses for all of its elements (see subginvcl 13087). (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 13071	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 12910	. . 3 class Grp
4	2	cv 1362	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1362	. . . . . 6 class s
7		cress 12480	. . . . . 6 class ↾s
8	4, 6, 7	co 5890	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2159	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 12479	. . . . . 6 class Base
11	4, 10	cfv 5230	. . . . 5 class ( Base ` w )
12	11	cpw 3589	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2471	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4078	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1363	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13077  subgex  13080