Theorem lmcls

Description: Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013) (Revised by Mario Carneiro, 1-May-2014)

	Ref	Expression
Hypotheses	lmff.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	lmff.3	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
	lmff.4	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	lmcls.5	⊢ ( 𝜑 → 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 )
	lmcls.7	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 )
	lmcls.8	⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 )
Assertion	lmcls	⊢ ( 𝜑 → 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		lmff.1	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		lmff.3	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
3		lmff.4	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
4		lmcls.5	⊢ ( 𝜑 → 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 )
5		lmcls.7	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 )
6		lmcls.8	⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 )
7	2 1 3	lmbr2	⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ↔ ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) )
8	4 7	mpbid	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ 𝑃 ∈ 𝑋 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) )
9	8	simp3d	⊢ ( 𝜑 → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) )
10	1	r19.2uz	⊢ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) → ∃ 𝑘 ∈ 𝑍 ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) )
11		inelcm	⊢ ( ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 ) → ( 𝑢 ∩ 𝑆 ) ≠ ∅ )
12	11	a1i	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑆 ) → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) )
13	5 12	mpan2d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) )
14	13	adantld	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) )
15	14	rexlimdva	⊢ ( 𝜑 → ( ∃ 𝑘 ∈ 𝑍 ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) )
16	10 15	syl5	⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) )
17	16	imim2d	⊢ ( 𝜑 → ( ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) → ( 𝑃 ∈ 𝑢 → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) ) )
18	17	ralimdv	⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) ) )
19	9 18	mpd	⊢ ( 𝜑 → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) )
20		topontop	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top )
21	2 20	syl	⊢ ( 𝜑 → 𝐽 ∈ Top )
22		toponuni	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )
23	2 22	syl	⊢ ( 𝜑 → 𝑋 = ∪ 𝐽 )
24	6 23	sseqtrd	⊢ ( 𝜑 → 𝑆 ⊆ ∪ 𝐽 )
25		lmcl	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑃 ) → 𝑃 ∈ 𝑋 )
26	2 4 25	syl2anc	⊢ ( 𝜑 → 𝑃 ∈ 𝑋 )
27	26 23	eleqtrd	⊢ ( 𝜑 → 𝑃 ∈ ∪ 𝐽 )
28		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
29	28	elcls	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽 ) → ( 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↔ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) ) )
30	21 24 27 29	syl3anc	⊢ ( 𝜑 → ( 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↔ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ( 𝑢 ∩ 𝑆 ) ≠ ∅ ) ) )
31	19 30	mpbird	⊢ ( 𝜑 → 𝑃 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )