Theorem ntrf2

Description: The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021)

	Ref	Expression
Hypotheses	ntrrn.x	⊢ 𝑋 = ∪ 𝐽
	ntrrn.i	⊢ 𝐼 = ( int ‘ 𝐽 )
Assertion	ntrf2	⊢ ( 𝐽 ∈ Top → 𝐼 : 𝒫 𝑋 ⟶ 𝒫 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		ntrrn.x	⊢ 𝑋 = ∪ 𝐽
2		ntrrn.i	⊢ 𝐼 = ( int ‘ 𝐽 )
3	1 2	ntrf	⊢ ( 𝐽 ∈ Top → 𝐼 : 𝒫 𝑋 ⟶ 𝐽 )
4	1	toptopon	⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
5		topgele	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( { ∅ , 𝑋 } ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋 ) )
6	4 5	sylbi	⊢ ( 𝐽 ∈ Top → ( { ∅ , 𝑋 } ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋 ) )
7	6	simprd	⊢ ( 𝐽 ∈ Top → 𝐽 ⊆ 𝒫 𝑋 )
8	3 7	fssd	⊢ ( 𝐽 ∈ Top → 𝐼 : 𝒫 𝑋 ⟶ 𝒫 𝑋 )