Theorem subrngss

Description: A subring is a subset. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypothesis	subrngss.1	$$⊢B={\mathrm{Base}}_R$$
Assertion	subrngss	$$⊢A\in \mathrm{SubRng}\left(R\right)\to A\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		subrngss.1	$$⊢B={\mathrm{Base}}_R$$
2	1	issubrng	$$⊢A\in \mathrm{SubRng}\left(R\right)\leftrightarrow \left(R\in \mathrm{Rng}\wedge R{\upharpoonright }_{𝑠}A\in \mathrm{Rng}\wedge A\subseteq B\right)$$
3	2	simp3bi	$$⊢A\in \mathrm{SubRng}\left(R\right)\to A\subseteq B$$