Metamath Proof Explorer


Theorem unicld

Description: A finite union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015)

Ref Expression
Hypothesis clscld.1 𝑋 = 𝐽
Assertion unicld ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ ( Clsd ‘ 𝐽 ) ) → 𝐴 ∈ ( Clsd ‘ 𝐽 ) )

Proof

Step Hyp Ref Expression
1 clscld.1 𝑋 = 𝐽
2 uniiun 𝐴 = 𝑥𝐴 𝑥
3 dfss3 ( 𝐴 ⊆ ( Clsd ‘ 𝐽 ) ↔ ∀ 𝑥𝐴 𝑥 ∈ ( Clsd ‘ 𝐽 ) )
4 1 iuncld ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀ 𝑥𝐴 𝑥 ∈ ( Clsd ‘ 𝐽 ) ) → 𝑥𝐴 𝑥 ∈ ( Clsd ‘ 𝐽 ) )
5 3 4 syl3an3b ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ ( Clsd ‘ 𝐽 ) ) → 𝑥𝐴 𝑥 ∈ ( Clsd ‘ 𝐽 ) )
6 2 5 eqeltrid ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ ( Clsd ‘ 𝐽 ) ) → 𝐴 ∈ ( Clsd ‘ 𝐽 ) )