Theorem cnfldsrngbas

Description: The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020)

	Ref	Expression
Hypothesis	cnfldsrngbas.r	⊢ 𝑅 = ( ℂfld ↾s 𝑆 )
Assertion	cnfldsrngbas	⊢ ( 𝑆 ⊆ ℂ → 𝑆 = ( Base ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		cnfldsrngbas.r	⊢ 𝑅 = ( ℂfld ↾s 𝑆 )
2		cnfldbas	⊢ ℂ = ( Base ‘ ℂfld )
3	1 2	ressbas2	⊢ ( 𝑆 ⊆ ℂ → 𝑆 = ( Base ‘ 𝑅 ) )