Metamath Proof Explorer

Theorem bracnvbra

Description: The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	bracnvbra	\|- ( T e. ( LinFn i^i ContFn ) -> ( bra ` ( `' bra ` T ) ) = T )

Step	Hyp	Ref	Expression
1		bra11	\|- bra : ~H -1-1-onto-> ( LinFn i^i ContFn )
2		f1ocnvfv2	\|- ( ( bra : ~H -1-1-onto-> ( LinFn i^i ContFn ) /\ T e. ( LinFn i^i ContFn ) ) -> ( bra ` ( `' bra ` T ) ) = T )
3	1 2	mpan	\|- ( T e. ( LinFn i^i ContFn ) -> ( bra ` ( `' bra ` T ) ) = T )

Step

Hyp

Ref

Expression

 |-  bra : ~H -1-1-onto-> ( LinFn i^i ContFn )

 |-  ( ( bra : ~H -1-1-onto-> ( LinFn i^i ContFn ) /\ T e. ( LinFn i^i ContFn ) ) -> ( bra ` ( `' bra ` T ) ) = T )

1 2

 |-  ( T e. ( LinFn i^i ContFn ) -> ( bra ` ( `' bra ` T ) ) = T )