Metamath Proof Explorer

Theorem cnvbracl

Description: Closure of the converse of the bra function. (Contributed by NM, 26-May-2006) (New usage is discouraged.)

		Ref	Expression
	Assertion	cnvbracl	\|- ( T e. ( LinFn i^i ContFn ) -> ( `' bra ` T ) e. ~H )

Step	Hyp	Ref	Expression
1		bra11	\|- bra : ~H -1-1-onto-> ( LinFn i^i ContFn )
2		f1ocnvdm	\|- ( ( bra : ~H -1-1-onto-> ( LinFn i^i ContFn ) /\ T e. ( LinFn i^i ContFn ) ) -> ( `' bra ` T ) e. ~H )
3	1 2	mpan	\|- ( T e. ( LinFn i^i ContFn ) -> ( `' bra ` T ) e. ~H )

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 ` T ) e. ~H )

1 2

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