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Metamath Proof Explorer

Theorem flfssfcf

Description: A limit point of a function is a cluster point of the function. (Contributed by Jeff Hankins, 28-Nov-2009) (Revised by Stefan O'Rear, 9-Aug-2015)

Ref Expression
Assertion flfssfcf ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ⊆ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 flimfcls ⊢ ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) ⊆ ( 𝐽 fClus ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) )
2 1 a1i ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) ⊆ ( 𝐽 fClus ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) )
3 flfval ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) = ( 𝐽 fLim ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) )
4 fcfval ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) = ( 𝐽 fClus ( ( 𝑋 FilMap 𝐹 ) ‘ 𝐿 ) ) )
5 2 3 4 3sstr4d ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ⊆ ( ( 𝐽 fClusf 𝐿 ) ‘ 𝐹 ) )