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Metamath Proof Explorer

Theorem fclsss1

Description: A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009) (Revised by Stefan O'Rear, 8-Aug-2015)

Ref Expression
Assertion fclsss1 ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) → ( 𝐾 fClus 𝐹 ) ⊆ ( 𝐽 fClus 𝐹 ) )

Proof

Step Hyp Ref Expression
1 simpl3 ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → 𝐽 ⊆ 𝐾 )
2 ssralv ⊢ ( 𝐽 ⊆ 𝐾 → ( ∀ 𝑜 ∈ 𝐾 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) → ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) )
3 2 anim2d ⊢ ( 𝐽 ⊆ 𝐾 → ( ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐾 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) → ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) )
4 1 3 syl ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → ( ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐾 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) → ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) )
5 simpl2 ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) )
6 fclstopon ⊢ ( 𝑥 ∈ ( 𝐾 fClus 𝐹 ) → ( 𝐾 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )
7 6 adantl ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → ( 𝐾 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )
8 5 7 mpbird ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → 𝐾 ∈ ( TopOn ‘ 𝑋 ) )
9 fclsopn ⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐾 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) )
10 8 5 9 syl2anc ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → ( 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐾 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) )
11 simpl1 ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
12 fclsopn ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) )
13 11 5 12 syl2anc ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) )
14 4 10 13 3imtr4d ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) ∧ 𝑥 ∈ ( 𝐾 fClus 𝐹 ) ) → ( 𝑥 ∈ ( 𝐾 fClus 𝐹 ) → 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) )
15 14 ex ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) → ( 𝑥 ∈ ( 𝐾 fClus 𝐹 ) → ( 𝑥 ∈ ( 𝐾 fClus 𝐹 ) → 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ) )
16 15 pm2.43d ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) → ( 𝑥 ∈ ( 𝐾 fClus 𝐹 ) → 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) )
17 16 ssrdv ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐽 ⊆ 𝐾 ) → ( 𝐾 fClus 𝐹 ) ⊆ ( 𝐽 fClus 𝐹 ) )