Theorem pjfoi

Description: A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjfn.1	|- H e. CH
Assertion	pjfoi	|- ( projh ` H ) : ~H -onto-> 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		df-fo	|- ( ( projh ` H ) : ~H -onto-> H <-> ( ( projh ` H ) Fn ~H /\ ran ( projh ` H ) = H ) )
5	2 3 4	mpbir2an	|- ( projh ` H ) : ~H -onto-> H