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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 ∈ ( 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 biimtrdi ⊢ ( ( 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 )