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 ) )