Theorem hlimuni

Description: A Hilbert space sequence converges to at most one limit. (Contributed by NM, 19-Aug-1999) (Revised by Mario Carneiro, 2-May-2015) (New usage is discouraged.)

	Ref	Expression
Assertion	hlimuni	$$⊢\left(F\underset{v}{⇝}A\wedge F\underset{v}{⇝}B\right)\to A=B$$

Proof

Step	Hyp	Ref	Expression
1		hlimf	$$⊢\underset{v}{⇝}:\mathrm{dom}\underset{v}{⇝}⟶\mathscr{H}$$
2		ffun	$$⊢\underset{v}{⇝}:\mathrm{dom}\underset{v}{⇝}⟶\mathscr{H}\to \mathrm{Fun}\underset{v}{⇝}$$
3		funbrfv	$$⊢\mathrm{Fun}\underset{v}{⇝}\to \left(F\underset{v}{⇝}A\to \underset{v}{⇝}\left(F\right)=A\right)$$
4	1 2 3	mp2b	$$⊢F\underset{v}{⇝}A\to \underset{v}{⇝}\left(F\right)=A$$
5		funbrfv	$$⊢\mathrm{Fun}\underset{v}{⇝}\to \left(F\underset{v}{⇝}B\to \underset{v}{⇝}\left(F\right)=B\right)$$
6	1 2 5	mp2b	$$⊢F\underset{v}{⇝}B\to \underset{v}{⇝}\left(F\right)=B$$
7	4 6	sylan9req	$$⊢\left(F\underset{v}{⇝}A\wedge F\underset{v}{⇝}B\right)\to A=B$$