Metamath Proof Explorer


Theorem lmcl

Description: Closure of a limit. (Contributed by NM, 19-Dec-2006) (Revised by Mario Carneiro, 23-Dec-2013)

Ref Expression
Assertion lmcl ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ( ⇝𝑡𝐽 ) 𝑃 ) → 𝑃𝑋 )

Proof

Step Hyp Ref Expression
1 id ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
2 1 lmbr ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝐹 ( ⇝𝑡𝐽 ) 𝑃 ↔ ( 𝐹 ∈ ( 𝑋pm ℂ ) ∧ 𝑃𝑋 ∧ ∀ 𝑢𝐽 ( 𝑃𝑢 → ∃ 𝑦 ∈ ran ℤ ( 𝐹𝑦 ) : 𝑦𝑢 ) ) ) )
3 2 biimpa ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ( ⇝𝑡𝐽 ) 𝑃 ) → ( 𝐹 ∈ ( 𝑋pm ℂ ) ∧ 𝑃𝑋 ∧ ∀ 𝑢𝐽 ( 𝑃𝑢 → ∃ 𝑦 ∈ ran ℤ ( 𝐹𝑦 ) : 𝑦𝑢 ) ) )
4 3 simp2d ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ( ⇝𝑡𝐽 ) 𝑃 ) → 𝑃𝑋 )