Theorem relssinxpdmrn

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

	Ref	Expression
Assertion	relssinxpdmrn	|- ( Rel R -> ( R C_ ( S i^i ( dom R X. ran R ) ) <-> R C_ S ) )

Proof

Step	Hyp	Ref	Expression
1		relssdmrn	|- ( Rel R -> R C_ ( dom R X. ran R ) )
2	1	biantrud	|- ( Rel R -> ( R C_ S <-> ( R C_ S /\ R C_ ( dom R X. ran R ) ) ) )
3		ssin	|- ( ( R C_ S /\ R C_ ( dom R X. ran R ) ) <-> R C_ ( S i^i ( dom R X. ran R ) ) )
4	2 3	bitr2di	|- ( Rel R -> ( R C_ ( S i^i ( dom R X. ran R ) ) <-> R C_ S ) )