Theorem pjini

Description: Membership of projection in an intersection. (Contributed by NM, 22-Apr-2001) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjocin.1	|- G e. CH
	pjocin.2	|- H e. CH
Assertion	pjini	|- ( A e. ( G i^i H ) -> ( ( projh ` G ) ` A ) e. ( G i^i H ) )

Proof

Step	Hyp	Ref	Expression
1		pjocin.1	|- G e. CH
2		pjocin.2	|- H e. CH
3		inss1	|- ( G i^i H ) C_ G
4	3	sseli	|- ( A e. ( G i^i H ) -> A e. G )
5		pjid	|- ( ( G e. CH /\ A e. G ) -> ( ( projh ` G ) ` A ) = A )
6	1 4 5	sylancr	|- ( A e. ( G i^i H ) -> ( ( projh ` G ) ` A ) = A )
7	6	eleq1d	|- ( A e. ( G i^i H ) -> ( ( ( projh ` G ) ` A ) e. ( G i^i H ) <-> A e. ( G i^i H ) ) )
8	7	ibir	|- ( A e. ( G i^i H ) -> ( ( projh ` G ) ` A ) e. ( G i^i H ) )