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 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ) → 𝑃 ∈ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
2	1	lmbr	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑦 ∈ ran ℤ≥ ( 𝐹 ↾ 𝑦 ) : 𝑦 ⟶ 𝑢 ) ) ) )
3	2	biimpa	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ) → ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑦 ∈ ran ℤ≥ ( 𝐹 ↾ 𝑦 ) : 𝑦 ⟶ 𝑢 ) ) )
4	3	simp2d	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ) → 𝑃 ∈ 𝑋 )