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 ∈ ( 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 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) )
4	1 2	ressbas2	⊢ ( 𝑆 ⊆ ( Base ‘ 𝐺 ) → 𝑆 = ( Base ‘ 𝐻 ) )
5	3 4	syl	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 = ( Base ‘ 𝐻 ) )