Theorem pjfi

Description: The mapping of a projection. (Contributed by NM, 11-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjfn.1	|- H e. CH
Assertion	pjfi	|- ( projh ` H ) : ~H --> ~H

Proof

Step	Hyp	Ref	Expression
1		pjfn.1	|- H e. CH
2	1	pjfni	|- ( projh ` H ) Fn ~H
3	1	pjrni	|- ran ( projh ` H ) = H
4	1	chssii	|- H C_ ~H
5	3 4	eqsstri	|- ran ( projh ` H ) C_ ~H
6		df-f	|- ( ( projh ` H ) : ~H --> ~H <-> ( ( projh ` H ) Fn ~H /\ ran ( projh ` H ) C_ ~H ) )
7	2 5 6	mpbir2an	|- ( projh ` H ) : ~H --> ~H