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	|- ( Rel B -> ( B C_ A <-> B C_ ( A |` _V ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-dfrel3	|- ( Rel B <-> ( B |` _V ) = B )
2		ssres	|- ( B C_ A -> ( B |` _V ) C_ ( A |` _V ) )
3		sseq1	|- ( ( B |` _V ) = B -> ( ( B |` _V ) C_ ( A |` _V ) <-> B C_ ( A |` _V ) ) )
4	2 3	imbitrid	|- ( ( B |` _V ) = B -> ( B C_ A -> B C_ ( A |` _V ) ) )
5		id	|- ( B C_ ( A |` _V ) -> B C_ ( A |` _V ) )
6		resss	|- ( A |` _V ) C_ A
7	5 6	sstrdi	|- ( B C_ ( A |` _V ) -> B C_ A )
8	4 7	impbid1	|- ( ( B |` _V ) = B -> ( B C_ A <-> B C_ ( A |` _V ) ) )
9	1 8	sylbi	|- ( Rel B -> ( B C_ A <-> B C_ ( A |` _V ) ) )