Theorem subgbas

Description: The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014)

	Ref	Expression
Hypothesis	subggrp.h	|- H = ( G |`s S )
Assertion	subgbas	|- ( S e. ( SubGrp ` G ) -> S = ( Base ` H ) )

Proof

Step	Hyp	Ref	Expression
1		subggrp.h	|- H = ( G |`s S )
2		eqid	|- ( Base ` G ) = ( Base ` G )
3	2	subgss	|- ( S e. ( SubGrp ` G ) -> S C_ ( Base ` G ) )
4	1 2	ressbas2	|- ( S C_ ( Base ` G ) -> S = ( Base ` H ) )
5	3 4	syl	|- ( S e. ( SubGrp ` G ) -> S = ( Base ` H ) )