Theorem resindm

Description: When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008)

	Ref	Expression
Assertion	resindm	|- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )

Proof

Step	Hyp	Ref	Expression
1		resdm	|- ( Rel A -> ( A |` dom A ) = A )
2	1	ineq2d	|- ( Rel A -> ( ( A |` B ) i^i ( A |` dom A ) ) = ( ( A |` B ) i^i A ) )
3		resindi	|- ( A |` ( B i^i dom A ) ) = ( ( A |` B ) i^i ( A |` dom A ) )
4		incom	|- ( ( A |` B ) i^i A ) = ( A i^i ( A |` B ) )
5		inres	|- ( A i^i ( A |` B ) ) = ( ( A i^i A ) |` B )
6		inidm	|- ( A i^i A ) = A
7	6	reseq1i	|- ( ( A i^i A ) |` B ) = ( A |` B )
8	4 5 7	3eqtrri	|- ( A |` B ) = ( ( A |` B ) i^i A )
9	2 3 8	3eqtr4g	|- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )