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Metamath Proof Explorer

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 ) )