Theorem hoeqi

Description: Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hoeq.1	|- S : ~H --> ~H
	hoeq.2	|- T : ~H --> ~H
Assertion	hoeqi	|- ( A. x e. ~H ( S ` x ) = ( T ` x ) <-> S = T )

Proof

Step	Hyp	Ref	Expression
1		hoeq.1	|- S : ~H --> ~H
2		hoeq.2	|- T : ~H --> ~H
3		hoeq	|- ( ( S : ~H --> ~H /\ T : ~H --> ~H ) -> ( A. x e. ~H ( S ` x ) = ( T ` x ) <-> S = T ) )
4	1 2 3	mp2an	|- ( A. x e. ~H ( S ` x ) = ( T ` x ) <-> S = T )