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Metamath Proof Explorer

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 )