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Metamath Proof Explorer

Theorem flimtop

Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015) (Revised by Stefan O'Rear, 9-Aug-2015)

Ref Expression
Assertion flimtop ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → 𝐽 ∈ Top )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ∪ 𝐽 = ∪ 𝐽
2 1 elflim2 ⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽 ) ∧ ( 𝐴 ∈ ∪ 𝐽 ∧ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ⊆ 𝐹 ) ) )
3 2 simplbi ⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽 ) )
4 3 simp1d ⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → 𝐽 ∈ Top )