Definition df-subg 15800

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 15818), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 15813), contains the neutral element of the group (see subg0 15809) and contains the inverses for all of its elements (see subginvcl 15812). (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 15797	. 2 class SubGrp
2		vw	. . 3 setvar w
3		cgrp 15532	. . 3 class Grp
4	2	cv 1369	. . . . . 6 class w
5		vs	. . . . . . 7 setvar s
6	5	cv 1369	. . . . . 6 class s
7		cress 14296	. . . . . 6 class ↾s
8	4, 6, 7	co 6203	. . . . 5 class ( w ↾s s )
9	8, 3	wcel 1758	. . . 4 wff ( w ↾s s ) e. Grp
10		cbs 14295	. . . . . 6 class Base
11	4, 10	cfv 5529	. . . . 5 class ( Base ` w )
12	11	cpw 3971	. . . 4 class ~P ( Base ` w )
13	9, 5, 12	crab 2803	. . 3 class { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp }
14	2, 3, 13	cmpt 4461	. 2 class ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )
15	1, 14	wceq 1370	1 wff SubGrp = ( w e. Grp |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Grp } )

This definition is referenced by:  issubg  15803