Theorem hausflf2

Description: If a convergent function has its values in a Hausdorff space, then it has a unique limit. (Contributed by FL, 14-Nov-2010) (Revised by Stefan O'Rear, 6-Aug-2015)

	Ref	Expression
Hypothesis	hausflf.x	|- X = U. J
Assertion	hausflf2	|- ( ( ( J e. Haus /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ ( ( J fLimf L ) ` F ) =/= (/) ) -> ( ( J fLimf L ) ` F ) ~~ 1o )

Proof

Step	Hyp	Ref	Expression
1		hausflf.x	|- X = U. J
2		n0	|- ( ( ( J fLimf L ) ` F ) =/= (/) <-> E. x x e. ( ( J fLimf L ) ` F ) )
3	2	biimpi	|- ( ( ( J fLimf L ) ` F ) =/= (/) -> E. x x e. ( ( J fLimf L ) ` F ) )
4	1	hausflf	|- ( ( J e. Haus /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> E* x x e. ( ( J fLimf L ) ` F ) )
5		euen1b	|- ( ( ( J fLimf L ) ` F ) ~~ 1o <-> E! x x e. ( ( J fLimf L ) ` F ) )
6		df-eu	|- ( E! x x e. ( ( J fLimf L ) ` F ) <-> ( E. x x e. ( ( J fLimf L ) ` F ) /\ E* x x e. ( ( J fLimf L ) ` F ) ) )
7	5 6	sylbbr	|- ( ( E. x x e. ( ( J fLimf L ) ` F ) /\ E* x x e. ( ( J fLimf L ) ` F ) ) -> ( ( J fLimf L ) ` F ) ~~ 1o )
8	3 4 7	syl2anr	|- ( ( ( J e. Haus /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ ( ( J fLimf L ) ` F ) =/= (/) ) -> ( ( J fLimf L ) ` F ) ~~ 1o )