Metamath Proof Explorer

Theorem fcfelbas

Description: A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009) (Revised by Stefan O'Rear, 9-Aug-2015)

		Ref	Expression
	Assertion	fcfelbas	$⊢ ((J \in TopOn (X) \land L \in Fil (Y) \land F : Y ⟶ X) \land A \in (J fClusf L) (F)) \to A \in X$

Proof

Step	Hyp	Ref	Expression
1		fcfval	$⊢ (J \in TopOn (X) \land L \in Fil (Y) \land F : Y ⟶ X) \to (J fClusf L) (F) = J fClus (X FilMap F) (L)$
2	1	eleq2d	$⊢ (J \in TopOn (X) \land L \in Fil (Y) \land F : Y ⟶ X) \to (A \in (J fClusf L) (F) \leftrightarrow A \in (J fClus (X FilMap F) (L)))$
3		eqid	$⊢ ⋃ J = ⋃ J$
4	3	fclselbas	$⊢ A \in (J fClus (X FilMap F) (L)) \to A \in ⋃ J$
5	2 4	syl6bi	$⊢ (J \in TopOn (X) \land L \in Fil (Y) \land F : Y ⟶ X) \to (A \in (J fClusf L) (F) \to A \in ⋃ J)$
6	5	imp	$⊢ ((J \in TopOn (X) \land L \in Fil (Y) \land F : Y ⟶ X) \land A \in (J fClusf L) (F)) \to A \in ⋃ J$
7		simpl1	$⊢ ((J \in TopOn (X) \land L \in Fil (Y) \land F : Y ⟶ X) \land A \in (J fClusf L) (F)) \to J \in TopOn (X)$
8		toponuni	$⊢ J \in TopOn (X) \to X = ⋃ J$
9	7 8	syl	$⊢ ((J \in TopOn (X) \land L \in Fil (Y) \land F : Y ⟶ X) \land A \in (J fClusf L) (F)) \to X = ⋃ J$
10	6 9	eleqtrrd	$⊢ ((J \in TopOn (X) \land L \in Fil (Y) \land F : Y ⟶ X) \land A \in (J fClusf L) (F)) \to A \in X$