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