Theorem subrgbas

Description: Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014)

	Ref	Expression
Hypothesis	subrgbas.b	|- S = ( R |`s A )
Assertion	subrgbas	|- ( A e. ( SubRing ` R ) -> A = ( Base ` S ) )

Proof

Step	Hyp	Ref	Expression
1		subrgbas.b	|- S = ( R |`s A )
2		subrgsubg	|- ( A e. ( SubRing ` R ) -> A e. ( SubGrp ` R ) )
3	1	subgbas	|- ( A e. ( SubGrp ` R ) -> A = ( Base ` S ) )
4	2 3	syl	|- ( A e. ( SubRing ` R ) -> A = ( Base ` S ) )