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 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) ) → 𝐴 ∈ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		fcfval	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) = ( 𝐽 fClus ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) )
2	1	eleq2d	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐴 ∈ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) ↔ 𝐴 ∈ ( 𝐽 fClus ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) ) )
3		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
4	3	fclselbas	⊢ ( 𝐴 ∈ ( 𝐽 fClus ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) → 𝐴 ∈ ∪ 𝐽 )
5	2 4	syl6bi	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐴 ∈ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) → 𝐴 ∈ ∪ 𝐽 ) )
6	5	imp	⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) ) → 𝐴 ∈ ∪ 𝐽 )
7		simpl1	⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
8		toponuni	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )
9	7 8	syl	⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) ) → 𝑋 = ∪ 𝐽 )
10	6 9	eleqtrrd	⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) ) → 𝐴 ∈ 𝑋 )