Definition df-subg 13702

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13721), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13716), contains the neutral element of the group (see subg0 13712) and contains the inverses for all of its elements (see subginvcl 13715). (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 13699	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13528	. . 3 class Grp
4	2	cv 1394	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1394	. . . . . 6 class s
7		cress 13028	. . . . . 6 class ↾s
8	4, 6, 7	co 6000	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2200	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13027	. . . . . 6 class Base
11	4, 10	cfv 5317	. . . . 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 4144	. 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  13705  subgex  13708