Definition df-subg 13950

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13969), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13964), contains the neutral element of the group (see subg0 13960) and contains the inverses for all of its elements (see subginvcl 13963). (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 13947	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 13782	. . 3 class Grp
4	2	cv 1401	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1401	. . . . . 6 class s
7		cress 13331	. . . . . 6 class ↾s
8	4, 6, 7	co 6075	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 2209	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 13330	. . . . . 6 class Base
11	4, 10	cfv 5372	. . . . 5 class ( Base ` w )
12	11	cpw 3685	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2532	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4187	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1402	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  13953  subgex  13956