Definition df-subg 15660

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 15678), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 15673), contains the neutral element of the group (see subg0 15669) and contains the inverses for all of its elements (see subginvcl 15672). (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 15657	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 15395	. . 3 class Grp
4	2	cv 1363	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1363	. . . . . 6 class s
7		cress 14160	. . . . . 6 class ↾s
8	4, 6, 7	co 6082	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1757	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 14159	. . . . . 6 class Base
11	4, 10	cfv 5408	. . . . 5 class ( Base ` w )
12	11	cpw 3850	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2711	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4340	. 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  15663