Theorem relssresv

Description: A relation is included in a class if and only if it is included in its "relation part" (see relpart ). (Contributed by BJ, 16-Oct-2026)

	Ref	Expression
Assertion	relssresv	$$⊢\mathrm{Rel}B\to \left(B\subseteq A\leftrightarrow B\subseteq {A\upharpoonright }_{\mathrm{V}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-dfrel3	$$⊢\mathrm{Rel}B\leftrightarrow {B\upharpoonright }_{\mathrm{V}}=B$$
2		ssres	$$⊢B\subseteq A\to {B\upharpoonright }_{\mathrm{V}}\subseteq {A\upharpoonright }_{\mathrm{V}}$$
3		sseq1	$$⊢{B\upharpoonright }_{\mathrm{V}}=B\to \left({B\upharpoonright }_{\mathrm{V}}\subseteq {A\upharpoonright }_{\mathrm{V}}\leftrightarrow B\subseteq {A\upharpoonright }_{\mathrm{V}}\right)$$
4	2 3	imbitrid	$$⊢{B\upharpoonright }_{\mathrm{V}}=B\to \left(B\subseteq A\to B\subseteq {A\upharpoonright }_{\mathrm{V}}\right)$$
5		id	$$⊢B\subseteq {A\upharpoonright }_{\mathrm{V}}\to B\subseteq {A\upharpoonright }_{\mathrm{V}}$$
6		resss	$$⊢{A\upharpoonright }_{\mathrm{V}}\subseteq A$$
7	5 6	sstrdi	$$⊢B\subseteq {A\upharpoonright }_{\mathrm{V}}\to B\subseteq A$$
8	4 7	impbid1	$$⊢{B\upharpoonright }_{\mathrm{V}}=B\to \left(B\subseteq A\leftrightarrow B\subseteq {A\upharpoonright }_{\mathrm{V}}\right)$$
9	1 8	sylbi	$$⊢\mathrm{Rel}B\to \left(B\subseteq A\leftrightarrow B\subseteq {A\upharpoonright }_{\mathrm{V}}\right)$$