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Metamath Proof Explorer

Theorem fclsss2

Description: A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009) (Revised by Mario Carneiro, 26-Aug-2015)

Ref Expression
Assertion fclsss2 ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝐽 fClus 𝐺 ) ⊆ ( 𝐽 fClus 𝐹 ) )

Proof

Step Hyp Ref Expression
1 simpl3 ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → 𝐹 ⊆ 𝐺 )
2 ssralv ⊢ ( 𝐹 ⊆ 𝐺 → ( ∀ 𝑠 ∈ 𝐺 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
3 1 2 syl ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → ( ∀ 𝑠 ∈ 𝐺 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
4 simpl1 ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
5 fclstopon ⊢ ( 𝑥 ∈ ( 𝐽 fClus 𝐺 ) → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) )
6 5 adantl ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) )
7 4 6 mpbid ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → 𝐺 ∈ ( Fil ‘ 𝑋 ) )
8 isfcls2 ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ↔ ∀ 𝑠 ∈ 𝐺 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
9 4 7 8 syl2anc ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ↔ ∀ 𝑠 ∈ 𝐺 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
10 simpl2 ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) )
11 isfcls2 ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ↔ ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
12 4 10 11 syl2anc ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ↔ ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
13 3 9 12 3imtr4d ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ∧ 𝑥 ∈ ( 𝐽 fClus 𝐺 ) ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐺 ) → 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) )
14 13 ex ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐺 ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐺 ) → 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) ) )
15 14 pm2.43d ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝑥 ∈ ( 𝐽 fClus 𝐺 ) → 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ) )
16 15 ssrdv ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) → ( 𝐽 fClus 𝐺 ) ⊆ ( 𝐽 fClus 𝐹 ) )