This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem hausflf

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

Ref Expression
Hypothesis hausflf.x ⊢ 𝑋 = ∪ 𝐽
Assertion hausflf ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ∃* 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 hausflf.x ⊢ 𝑋 = ∪ 𝐽
2 hausflimi ⊢ ( 𝐽 ∈ Haus → ∃* 𝑥 𝑥 ∈ ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) )
3 2 3ad2ant1 ⊢ ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ∃* 𝑥 𝑥 ∈ ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) )
4 haustop ⊢ ( 𝐽 ∈ Haus → 𝐽 ∈ Top )
5 1 toptopon ⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
6 4 5 sylib ⊢ ( 𝐽 ∈ Haus → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
7 flfval ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) = ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) )
8 6 7 syl3an1 ⊢ ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) = ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) )
9 8 eleq2d ⊢ ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ 𝑥 ∈ ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) ) )
10 9 mobidv ⊢ ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ∃* 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ∃* 𝑥 𝑥 ∈ ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) ) )
11 3 10 mpbird ⊢ ( ( 𝐽 ∈ Haus ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ∃* 𝑥 𝑥 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) )