Theorem pjidmi

Description: A projection is idempotent. Property (ii) of Beran p. 109. (Contributed by NM, 28-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjidm.1	|- H e. CH
	pjidm.2	|- A e. ~H
Assertion	pjidmi	|- ( ( projh ` H ) ` ( ( projh ` H ) ` A ) ) = ( ( projh ` H ) ` A )

Proof

Step	Hyp	Ref	Expression
1		pjidm.1	|- H e. CH
2		pjidm.2	|- A e. ~H
3	1 2	pjclii	|- ( ( projh ` H ) ` A ) e. H
4	1 2	pjhclii	|- ( ( projh ` H ) ` A ) e. ~H
5	1 4	pjchi	|- ( ( ( projh ` H ) ` A ) e. H <-> ( ( projh ` H ) ` ( ( projh ` H ) ` A ) ) = ( ( projh ` H ) ` A ) )
6	3 5	mpbi	|- ( ( projh ` H ) ` ( ( projh ` H ) ` A ) ) = ( ( projh ` H ) ` A )