Theorem flimtopon

Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015)

	Ref	Expression
Assertion	flimtopon	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		flimtop	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → 𝐽 ∈ Top )
2		istopon	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ ( 𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽 ) )
3	2	baib	⊢ ( 𝐽 ∈ Top → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝑋 = ∪ 𝐽 ) )
4	1 3	syl	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝑋 = ∪ 𝐽 ) )
5		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
6	5	flimfil	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) )
7		fveq2	⊢ ( 𝑋 = ∪ 𝐽 → ( Fil ‘ 𝑋 ) = ( Fil ‘ ∪ 𝐽 ) )
8	7	eleq2d	⊢ ( 𝑋 = ∪ 𝐽 → ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ↔ 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ) )
9	6 8	syl5ibrcom	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝑋 = ∪ 𝐽 → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )
10		filunibas	⊢ ( 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) → ∪ 𝐹 = ∪ 𝐽 )
11	6 10	syl	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ∪ 𝐹 = ∪ 𝐽 )
12		filunibas	⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ∪ 𝐹 = 𝑋 )
13	12	eqeq1d	⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽 ) )
14	11 13	syl5ibcom	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) )
15	9 14	impbid	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )
16	4 15	bitrd	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )