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 ) )