Theorem relssinxpdmrn

Description: Subset of restriction, special case. (Contributed by Peter Mazsa, 10-Apr-2023)

	Ref	Expression
Assertion	relssinxpdmrn	$$⊢\mathrm{Rel}R\to \left(R\subseteq S\cap \left(\mathrm{dom}R\times \mathrm{ran}R\right)\leftrightarrow R\subseteq S\right)$$

Proof

Step	Hyp	Ref	Expression
1		relssdmrn	$$⊢\mathrm{Rel}R\to R\subseteq \mathrm{dom}R\times \mathrm{ran}R$$
2	1	biantrud	$$⊢\mathrm{Rel}R\to \left(R\subseteq S\leftrightarrow \left(R\subseteq S\wedge R\subseteq \mathrm{dom}R\times \mathrm{ran}R\right)\right)$$
3		ssin	$$⊢\left(R\subseteq S\wedge R\subseteq \mathrm{dom}R\times \mathrm{ran}R\right)\leftrightarrow R\subseteq S\cap \left(\mathrm{dom}R\times \mathrm{ran}R\right)$$
4	2 3	bitr2di	$$⊢\mathrm{Rel}R\to \left(R\subseteq S\cap \left(\mathrm{dom}R\times \mathrm{ran}R\right)\leftrightarrow R\subseteq S\right)$$