Theorem ntrclskex

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-May-2021)

	Ref	Expression
Hypotheses	ntrcls.o	$$⊢O=\left(i\in \mathrm{V}⟼\left(k\in \left({𝒫i}^{𝒫i}\right)⟼\left(j\in 𝒫i⟼i\setminus k\left(i\setminus j\right)\right)\right)\right)$$
	ntrcls.d	$$⊢D=O\left(B\right)$$
	ntrcls.r	$$⊢\phi \to IDK$$
Assertion	ntrclskex	$$⊢\phi \to K\in \left({𝒫B}^{𝒫B}\right)$$

Proof

Step	Hyp	Ref	Expression
1		ntrcls.o	$$⊢O=\left(i\in \mathrm{V}⟼\left(k\in \left({𝒫i}^{𝒫i}\right)⟼\left(j\in 𝒫i⟼i\setminus k\left(i\setminus j\right)\right)\right)\right)$$
2		ntrcls.d	$$⊢D=O\left(B\right)$$
3		ntrcls.r	$$⊢\phi \to IDK$$
4	1 2 3	ntrclsnvobr	$$⊢\phi \to KDI$$
5	1 2 4	ntrclsiex	$$⊢\phi \to K\in \left({𝒫B}^{𝒫B}\right)$$