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 = ∪ 𝐽
Assertion	hausflf2	⊢ ( ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ≠ ∅ ) → ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ≈ 1o )

Proof

Step	Hyp	Ref	Expression
1		hausflf.x	⊢ 𝑋 = ∪ 𝐽
2		n0	⊢ ( ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) )
3	2	biimpi	⊢ ( ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ≠ ∅ → ∃ 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) )
4	1	hausflf	⊢ ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ∃* 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) )
5		euen1b	⊢ ( ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ≈ 1o ↔ ∃! 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) )
6		df-eu	⊢ ( ∃! 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ( ∃ 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ∧ ∃* 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ) )
7	5 6	sylbbr	⊢ ( ( ∃ 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ∧ ∃* 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ) → ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ≈ 1o )
8	3 4 7	syl2anr	⊢ ( ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ≠ ∅ ) → ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ≈ 1o )