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 ∈ ( LinFn ∩ ContFn ) → ( bra ‘ ( ◡ bra ‘ 𝑇 ) ) = 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		bra11	⊢ bra : ℋ –1-1-onto→ ( LinFn ∩ ContFn )
2		f1ocnvfv2	⊢ ( ( bra : ℋ –1-1-onto→ ( LinFn ∩ ContFn ) ∧ 𝑇 ∈ ( LinFn ∩ ContFn ) ) → ( bra ‘ ( ◡ bra ‘ 𝑇 ) ) = 𝑇 )
3	1 2	mpan	⊢ ( 𝑇 ∈ ( LinFn ∩ ContFn ) → ( bra ‘ ( ◡ bra ‘ 𝑇 ) ) = 𝑇 )