P. 451 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45001-45100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	clsk1indlem2 45001*	The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)
⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3o𝑠 ⊆ (𝐾‘𝑠)
Theorem	clsk1indlem3 45002*	The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K3 property of being sub-linear. (Contributed by RP, 6-Jul-2021.)
⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3o∀𝑡 ∈ 𝒫 3o(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))
Theorem	clsk1indlem4 45003*	The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.)
⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)
Theorem	clsk1indlem1 45004*	The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.)
⊢ 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))    ⇒   ⊢ ∃𝑠 ∈ 𝒫 3o∃𝑡 ∈ 𝒫 3o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))
Theorem	clsk1independent 45005*	For generalized closure functions, property K1 (isotony) is independent of the properties K0, K2, K3, K4. This contradicts a claim which appears in preprints of Table 2 in Bärbel M. R. Stadler and Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci., 42:577-585, 2002. Proceedings MCC 2001, Dubrovnik. The same table row implying K1 follows from the other four appears in the supplemental materials Bärbel M. R. Stadler and Peter F. Stadler. "Basic Properties of Closure Spaces" 2001 on page 12. (Contributed by RP, 5-Jul-2021.)
⊢ (𝜑 ↔ (𝑘‘∅) = ∅)    &   ⊢ (𝜓 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑠 ⊆ 𝑡 → (𝑘‘𝑠) ⊆ (𝑘‘𝑡)))    &   ⊢ (𝜒 ↔ ∀𝑠 ∈ 𝒫 𝑏𝑠 ⊆ (𝑘‘𝑠))    &   ⊢ (𝜃 ↔ ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)))    &   ⊢ (𝜏 ↔ ∀𝑠 ∈ 𝒫 𝑏(𝑘‘(𝑘‘𝑠)) = (𝑘‘𝑠))    ⇒   ⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏)(((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜓)
Theorem	neik0pk1imk0 45006*	Kuratowski's K0' and K1 axioms imply K0. Neighborhood version. (Contributed by RP, 3-Jun-2021.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥)))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥))
Theorem	isotone1 45007*	Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)
⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))
Theorem	isotone2 45008*	Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)
⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝐹‘(𝑎 ∩ 𝑏)) ⊆ ((𝐹‘𝑎) ∩ (𝐹‘𝑏)))
Theorem	ntrk1k3eqk13 45009*	An interior function is both monotone and sub-linear if and only if it is finitely linear. (Contributed by RP, 18-Jun-2021.)
⊢ ((∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡))) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)))
Theorem	ntrclsf1o 45010*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
Theorem	ntrclsnvobr 45011*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then they are related the opposite way. (Contributed by RP, 21-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐾𝐷𝐼)
Theorem	ntrclsiex 45012*	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.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
Theorem	ntrclskex 45013*	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.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
Theorem	ntrclsfv1 45014*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
Theorem	ntrclsfv2 45015*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼)
Theorem	ntrclselnel1 45016*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 28-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))
Theorem	ntrclselnel2 45017*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in interior of the complement of a set and non-membership in the closure of the set. (Contributed by RP, 28-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑆)))
Theorem	ntrclsfv 45018*	The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
Theorem	ntrclsfveq1 45019*	If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶)))
Theorem	ntrclsfveq2 45020*	If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶)))
Theorem	ntrclsfveq 45021*	If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))
Theorem	ntrclsss 45022*	If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆))))
Theorem	ntrclsneine0lem 45023*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that at least one (pseudo-)neighborbood of a particular point exists hold equally. (Contributed by RP, 21-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠)))
Theorem	ntrclsneine0 45024*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 21-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐾‘𝑠)))
Theorem	ntrclscls00 45025*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Theorem	ntrclsiso 45026*	If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡))))
Theorem	ntrclsk2 45027*	An interior function is contracting if and only if the closure function is expansive. (Contributed by RP, 9-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))
Theorem	ntrclskb 45028*	The interiors of disjoint sets are disjoint if and only if the closures of sets that span the base set also span the base set. (Contributed by RP, 10-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = 𝐵)))
Theorem	ntrclsk3 45029*	The intersection of interiors of a every pair is a subset of the interior of the intersection of the pair if an only if the closure of the union of every pair is a subset of the union of closures of the pair. (Contributed by RP, 19-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
Theorem	ntrclsk13 45030*	The interior of the intersection of any pair is equal to the intersection of the interiors if and only if the closure of the unions of any pair is equal to the union of closures. (Contributed by RP, 19-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))))
Theorem	ntrclsk4 45031*	Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-2021.)
⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
Theorem	ntrneibex 45032*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the base set exists. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)
Theorem	ntrneircomplex 45033*	The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
Theorem	ntrneif1o 45034*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	ntrneiiex 45035*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the interior function exists. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
Theorem	ntrneinex 45036*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the neighborhood function exists. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	ntrneicnv 45037*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then converse of 𝐹 is known. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → ◡𝐹 = (𝐵𝑂𝒫 𝐵))
Theorem	ntrneifv1 45038*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of 𝐹 is the neighborhood function. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)
Theorem	ntrneifv2 45039*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of converse of 𝐹 is the interior function. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (◡𝐹‘𝑁) = 𝐼)
Theorem	ntrneiel 45040*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋)))
Theorem	ntrneifv3 45041*	The value of the neighbors (convergents) expressed in terms of the interior (closure) function. (Contributed by RP, 26-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)})
Theorem	ntrneineine0lem 45042*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ ∅))
Theorem	ntrneineine1lem 45043*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ 𝒫 𝐵))
Theorem	ntrneifv4 45044*	The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)})
Theorem	ntrneiel2 45045*	Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (Contributed by RP, 11-Jul-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐼‘𝑆)) ↔ ∃𝑢 ∈ (𝑁‘𝑋)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑆 ∈ (𝑁‘𝑦))))
Theorem	ntrneineine0 45046*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅))
Theorem	ntrneineine1 45047*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ 𝒫 𝐵))
Theorem	ntrneicls00 45048*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥)))
Theorem	ntrneicls11 45049*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ ∅ ∈ (𝑁‘𝑥)))
Theorem	ntrneiiso 45050*	If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior function is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))))
Theorem	ntrneik2 45051*	An interior function is contracting if and only if all the neighborhoods of a point contain that point. (Contributed by RP, 11-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠)))
Theorem	ntrneix2 45052*	An interior (closure) function is expansive if and only if all subsets which contain a point are neighborhoods (convergents) of that point. (Contributed by RP, 11-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑁‘𝑥))))
Theorem	ntrneikb 45053*	The interiors of disjoint sets are disjoint if and only if the neighborhoods of every point contain no disjoint sets. (Contributed by RP, 11-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ≠ ∅)))
Theorem	ntrneixb 45054*	The interiors (closures) of sets that span the base set also span the base set if and only if the neighborhoods (convergents) of every point contain at least one of every pair of sets that span the base set. (Contributed by RP, 11-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
Theorem	ntrneik3 45055*	The intersection of interiors of any pair is a subset of the interior of the intersection if and only if the intersection of any two neighborhoods of a point is also a neighborhood. (Contributed by RP, 19-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥))))
Theorem	ntrneix3 45056*	The closure of the union of any pair is a subset of the union of closures if and only if the union of any pair belonging to the convergents of a point implies at least one of the pair belongs to the the convergents of that point. (Contributed by RP, 19-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
Theorem	ntrneik13 45057*	The interior of the intersection of any pair equals intersection of interiors if and only if the intersection of any pair belonging to the neighborhood of a point is equivalent to both of the pair belonging to the neighborhood of that point. (Contributed by RP, 19-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)))))
Theorem	ntrneix13 45058*	The closure of the union of any pair is equal to the union of closures if and only if the union of any pair belonging to the convergents of a point if equivalent to at least one of the pain belonging to the convergents of that point. (Contributed by RP, 19-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠 ∪ 𝑡)) = ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
Theorem	ntrneik4w 45059*	Idempotence of the interior function is equivalent to saying a set is a neighborhood of a point if and only if the interior of the set is a neighborhood of a point. (Contributed by RP, 11-Jul-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))
Theorem	ntrneik4 45060*	Idempotence of the interior function is equivalent to stating a set, 𝑠, is a neighborhood of a point, 𝑥 is equivalent to there existing a special neighborhood, 𝑢, of 𝑥 such that a point is an element of the special neighborhood if and only if 𝑠 is also a neighborhood of the point. (Contributed by RP, 11-Jul-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ (𝜑 → 𝐼𝐹𝑁)    ⇒   ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))))
Theorem	clsneibex 45061	If (pseudo-)closure and (pseudo-)neighborhood functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)
⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)
Theorem	clsneircomplex 45062	The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
Theorem	clsneif1o 45063*	If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the operator is a one-to-one, onto mapping. (Contributed by RP, 5-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → 𝐻:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	clsneicnv 45064*	If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the converse of the operator is known. (Contributed by RP, 5-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → ◡𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
Theorem	clsneikex 45065*	If closure and neighborhoods functions are related, the closure function exists. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
Theorem	clsneinex 45066*	If closure and neighborhoods functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    ⇒   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	clsneiel1 45067*	If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of a subset is equivalent to the complement of the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
Theorem	clsneiel2 45068*	If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of the complement of a subset is equivalent to the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑆 ∈ (𝑁‘𝑋)))
Theorem	clsneifv3 45069*	Value of the neighborhoods (convergents) in terms of the closure (interior) function. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑠))})
Theorem	clsneifv4 45070*	Value of the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)})
Theorem	neicvgbex 45071	If (pseudo-)neighborhood and (pseudo-)convergent functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)
⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)
Theorem	neicvgrcomplex 45072	The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
Theorem	neicvgf1o 45073*	If neighborhood and convergent functions are related by operator 𝐻, it is a one-to-one onto relation. (Contributed by RP, 11-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝐻:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	neicvgnvo 45074*	If neighborhood and convergent functions are related by operator 𝐻, it is its own converse function. (Contributed by RP, 11-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → ◡𝐻 = 𝐻)
Theorem	neicvgnvor 45075*	If neighborhood and convergent functions are related by operator 𝐻, the relationship holds with the functions swapped. (Contributed by RP, 11-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝑀𝐻𝑁)
Theorem	neicvgmex 45076*	If the neighborhoods and convergents functions are related, the convergents function exists. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝑀 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	neicvgnex 45077*	If the neighborhoods and convergents functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	neicvgel1 45078*	A subset being an element of a neighborhood of a point is equivalent to the complement of that subset not being a element of the convergent of that point. (Contributed by RP, 12-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑆 ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑋)))
Theorem	neicvgel2 45079*	The complement of a subset being an element of a neighborhood at a point is equivalent to that subset not being a element of the convergent at that point. (Contributed by RP, 12-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋)))
Theorem	neicvgfv 45080*	The value of the neighborhoods (convergents) in terms of the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)})
Theorem	ntrrn 45081	The range of the interior function of a topology a subset of the open sets of the topology. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐼 = (int‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽)
Theorem	ntrf 45082	The interior function of a topology is a map from the powerset of the base set to the open sets of the topology. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐼 = (int‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽)
Theorem	ntrf2 45083	The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐼 = (int‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋)
Theorem	ntrelmap 45084	The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐼 = (int‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
Theorem	clsf2 45085	The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 23346. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)
Theorem	clselmap 45086	The closure function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
Theorem	dssmapntrcls 45087*	The interior and closure operators on a topology are duals of each other. See also kur14lem2 35941. (Contributed by RP, 21-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝑋)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾))
Theorem	dssmapclsntr 45088*	The closure and interior operators on a topology are duals of each other. See also kur14lem2 35941. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝑋)    ⇒   ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼))
21.38.7.3  Generic Neighborhood Spaces Any neighborhood space is an open set topology and any open set topology is a neighborhood space. Seifert and Threlfall define a generic neighborhood space which is a superset of what is now generally used and related concepts and the following will show that those definitions apply to elements of Top. Seifert and Threlfall do not allow neighborhood spaces on the empty set while sn0top 23297 is an example of a topology with an empty base set. This divergence is unlikely to pose serious problems.
Theorem	gneispa 45089*	Each point 𝑝 of the neighborhood space has at least one neighborhood; each neighborhood of 𝑝 contains 𝑝. Axiom A of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))
Theorem	gneispb 45090*	Given a neighborhood 𝑁 of 𝑃, each subset of the neighborhood space containing this neighborhood is also a neighborhood of 𝑃. Axiom B of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁 ⊆ 𝑠 → 𝑠 ∈ ((nei‘𝐽)‘{𝑃})))
Theorem	gneispace2 45091*	The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
Theorem	gneispace3 45092*	The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
Theorem	gneispace 45093*	The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (Contributed by RP, 14-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))))
Theorem	gneispacef 45094*	A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
Theorem	gneispacef2 45095*	A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)
Theorem	gneispacefun 45096*	A generic neighborhood space is a function. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → Fun 𝐹)
Theorem	gneispacern 45097*	A generic neighborhood space has a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
Theorem	gneispacern2 45098*	A generic neighborhood space has a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹)
Theorem	gneispace0nelrn 45099*	A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
Theorem	gneispace0nelrn2 45100*	A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) → (𝐹‘𝑃) ≠ ∅)