Theorem neircl

Description: Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024)

	Ref	Expression
Assertion	neircl	$$⊢N\in \mathrm{nei}\left(J\right)\left(S\right)\to J\in \mathrm{Top}$$

Proof

Step	Hyp	Ref	Expression
1		elfvne0	$$⊢N\in \mathrm{nei}\left(J\right)\left(S\right)\to \mathrm{nei}\left(J\right)\ne \varnothing$$
2		n0	$$⊢\mathrm{nei}\left(J\right)\ne \varnothing \leftrightarrow \exists ff\in \mathrm{nei}\left(J\right)$$
3	2	biimpi	$$⊢\mathrm{nei}\left(J\right)\ne \varnothing \to \exists ff\in \mathrm{nei}\left(J\right)$$
4		df-nei	$$⊢\mathrm{nei}=\left(j\in \mathrm{Top}⟼\left(x\in 𝒫\bigcup j⟼\left\{y\in 𝒫\bigcup j|\exists g\in j\left(x\subseteq g\wedge g\subseteq y\right)\right\}\right)\right)$$
5	4	mptrcl	$$⊢f\in \mathrm{nei}\left(J\right)\to J\in \mathrm{Top}$$
6	5	exlimiv	$$⊢\exists ff\in \mathrm{nei}\left(J\right)\to J\in \mathrm{Top}$$
7	1 3 6	3syl	$$⊢N\in \mathrm{nei}\left(J\right)\left(S\right)\to J\in \mathrm{Top}$$