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Theorem List for Metamath Proof Explorer - 40301-40400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremntrclsk13 40301* 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 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐾‘(𝑠𝑡)) = ((𝐾𝑠) ∪ (𝐾𝑡))))
 
Theoremntrclsk4 40302* Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-2021.)
𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))    &   𝐷 = (𝑂𝐵)    &   (𝜑𝐼𝐷𝐾)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾𝑠)) = (𝐾𝑠)))
 
Theoremntrneibex 40303* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝐵 ∈ V)
 
Theoremntrneircomplex 40304* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
 
Theoremntrneif1o 40305* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
 
Theoremntrneiiex 40306* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
 
Theoremntrneinex 40307* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
 
Theoremntrneicnv 40308* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑𝐹 = (𝐵𝑂𝒫 𝐵))
 
Theoremntrneifv1 40309* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (𝐹𝐼) = 𝑁)
 
Theoremntrneifv2 40310* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (𝐹𝑁) = 𝐼)
 
Theoremntrneiel 40311* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))
 
Theoremntrneifv3 40312* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)})
 
Theoremntrneineine0lem 40313* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ ∅))
 
Theoremntrneineine1lem 40314* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))
 
Theoremntrneifv4 40315* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝐼𝑆) = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
 
Theoremntrneiel2 40316* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
 
Theoremntrneineine0 40317* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵 (𝑁𝑥) ≠ ∅))
 
Theoremntrneineine1 40318* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵 (𝑁𝑥) ≠ 𝒫 𝐵))
 
Theoremntrneicls00 40319* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))
 
Theoremntrneicls11 40320* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
 
Theoremntrneiiso 40321* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
 
Theoremntrneik2 40322* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) → 𝑥𝑠)))
 
Theoremntrneix2 40323* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑥𝑠𝑠 ∈ (𝑁𝑥))))
 
Theoremntrneikb 40324* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ≠ ∅)))
 
Theoremntrneixb 40325* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
 
Theoremntrneik3 40326* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
 
Theoremntrneix3 40327* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
 
Theoremntrneik13 40328* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))
 
Theoremntrneix13 40329* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
 
Theoremntrneik4w 40330* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ (𝐼𝑠) ∈ (𝑁𝑥))))
 
Theoremntrneik4 40331* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦)))))
 
Theoremclsneibex 40332 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 = (𝑃𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐵 ∈ V)
 
Theoremclsneircomplex 40333 The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
 
Theoremclsneif1o 40334* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐻:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
 
Theoremclsneicnv 40335* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))
 
Theoremclsneikex 40336* If closure and neighborhoods functions are related, the closure function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
 
Theoremclsneinex 40337* If closure and neighborhoods functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
 
Theoremclsneiel1 40338* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))
 
Theoremclsneiel2 40339* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐾‘(𝐵𝑆)) ↔ ¬ 𝑆 ∈ (𝑁𝑋)))
 
Theoremclsneifv3 40340* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑠))})
 
Theoremclsneifv4 40341* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝐾𝑆) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})
 
Theoremneicvgbex 40342 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 = (𝑃𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐵 ∈ V)
 
Theoremneicvgrcomplex 40343 The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
 
Theoremneicvgf1o 40344* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐻:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
 
Theoremneicvgnvo 40345* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐻 = 𝐻)
 
Theoremneicvgnvor 40346* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑀𝐻𝑁)
 
Theoremneicvgmex 40347* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑀 ∈ (𝒫 𝒫 𝐵m 𝐵))
 
Theoremneicvgnex 40348* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
 
Theoremneicvgel1 40349* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑆 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))
 
Theoremneicvgel2 40350* 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 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → ((𝐵𝑆) ∈ (𝑁𝑋) ↔ ¬ 𝑆 ∈ (𝑀𝑋)))
 
Theoremneicvgfv 40351* The value of the neighborhoods (convergents) in terms of the the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})
 
Theoremntrrn 40352 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 𝐼𝐽)
 
Theoremntrf 40353 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 𝑋𝐽)
 
Theoremntrf2 40354 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)
 
Theoremntrelmap 40355 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 𝑋m 𝒫 𝑋))
 
Theoremclsf2 40356 The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 21586. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)       (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)
 
Theoremclselmap 40357 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 𝑋m 𝒫 𝑋))
 
Theoremdssmapntrcls 40358* The interior and closure operators on a topology are duals of each other. See also kur14lem2 32352. (Contributed by RP, 21-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝑋)       (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
 
Theoremdssmapclsntr 40359* The closure and interior operators on a topology are duals of each other. See also kur14lem2 32352. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝑋)       (𝐽 ∈ Top → 𝐾 = (𝐷𝐼))
 
20.30.4.3  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 21537 is an example of a topology with an empty base set. This divergence is unlikely to pose serious problems.

 
Theoremgneispa 40360* 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 → ∀𝑝𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
 
Theoremgneispb 40361* 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 ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁𝑠𝑠 ∈ ((nei‘𝐽)‘{𝑃})))
 
Theoremgneispace2 40362* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))
 
Theoremgneispace3 40363* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))
 
Theoremgneispace 40364* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))))
 
Theoremgneispacef 40365* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
 
Theoremgneispacef2 40366* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)
 
Theoremgneispacefun 40367* A generic neighborhood space is a function. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → Fun 𝐹)
 
Theoremgneispacern 40368* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
 
Theoremgneispacern2 40369* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹)
 
Theoremgneispace0nelrn 40370* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹𝑝) ≠ ∅)
 
Theoremgneispace0nelrn2 40371* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       ((𝐹𝐴𝑃 ∈ dom 𝐹) → (𝐹𝑃) ≠ ∅)
 
Theoremgneispace0nelrn3 40372* 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 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ¬ ∅ ∈ ran 𝐹)
 
Theoremgneispaceel 40373* Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)𝑝𝑛)
 
Theoremgneispaceel2 40374* Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       ((𝐹𝐴𝑃 ∈ dom 𝐹𝑁 ∈ (𝐹𝑃)) → 𝑃𝑁)
 
Theoremgneispacess 40375* All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
 
Theoremgneispacess2 40376* All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (((𝐹𝐴𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆)) → 𝑆 ∈ (𝐹𝑃))
 
20.30.5  Exploring Higher Homotopy via Kerodon

See https://kerodon.net/ for a work in progress by Jacob Lurie.

 
20.30.5.1  Simplicial Sets

See https://kerodon.net/tag/0004 for introduction to the topological simplex of dimension 𝑁.

 
Theoremk0004lem1 40377 Application of ssin 4206 to range of a function. (Contributed by RP, 1-Apr-2021.)
(𝐷 = (𝐵𝐶) → ((𝐹:𝐴𝐵 ∧ (𝐹𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴𝐷))
 
Theoremk0004lem2 40378 A mapping with a particular restricted range is also a mapping to that range. (Contributed by RP, 1-Apr-2021.)
((𝐴𝑈𝐵𝑉𝐶𝐵) → ((𝐹 ∈ (𝐵m 𝐴) ∧ (𝐹𝐴) ⊆ 𝐶) ↔ 𝐹 ∈ (𝐶m 𝐴)))
 
Theoremk0004lem3 40379 When the value of a mapping on a singleton is known, the mapping is a completely known singleton. (Contributed by RP, 2-Apr-2021.)
((𝐴𝑈𝐵𝑉𝐶𝐵) → ((𝐹 ∈ (𝐵m {𝐴}) ∧ (𝐹𝐴) = 𝐶) ↔ 𝐹 = {⟨𝐴, 𝐶⟩}))
 
Theoremk0004val 40380* The topological simplex of dimension 𝑁 is the set of real vectors where the components are nonnegative and sum to 1. (Contributed by RP, 29-Mar-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝑁 ∈ ℕ0 → (𝐴𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡𝑘) = 1})
 
Theoremk0004ss1 40381* The topological simplex of dimension 𝑁 is a subset of the real vectors of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝑁 ∈ ℕ0 → (𝐴𝑁) ⊆ (ℝ ↑m (1...(𝑁 + 1))))
 
Theoremk0004ss2 40382* The topological simplex of dimension 𝑁 is a subset of the base set of a real vector space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝑁 ∈ ℕ0 → (𝐴𝑁) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1)))))
 
Theoremk0004ss3 40383* The topological simplex of dimension 𝑁 is a subset of the base set of Euclidean space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝑁 ∈ ℕ0 → (𝐴𝑁) ⊆ (Base‘(𝔼hil‘(𝑁 + 1))))
 
Theoremk0004val0 40384* The topological simplex of dimension 0 is a singleton. (Contributed by RP, 2-Apr-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑m (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝐴‘0) = {{⟨1, 1⟩}}
 
20.31  Mathbox for Stanislas Polu
 
Theoreminductionexd 40385 Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))
 
20.31.1  IMO Problems
 
20.31.1.1  IMO 1972 B2
 
Theoremwwlemuld 40386 Natural deduction form of lemul2d 12465. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))    &   (𝜑 → 0 < 𝐶)       (𝜑𝐴𝐵)
 
Theoremleeq1d 40387 Specialization of breq1d 5068 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴𝐶)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑𝐵𝐶)
 
Theoremleeq2d 40388 Specialization of breq2d 5070 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴𝐶)    &   (𝜑𝐶 = 𝐷)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑𝐴𝐷)
 
Theoremabsmulrposd 40389 Specialization of absmuld with absidd 14772. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (abs‘(𝐴 · 𝐵)) = (𝐴 · (abs‘𝐵)))
 
Theoremimadisjld 40390 Natural dduction form of one side of imadisj 5942. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑 → (dom 𝐴𝐵) = ∅)       (𝜑 → (𝐴𝐵) = ∅)
 
Theoremimadisjlnd 40391 Natural deduction form of one negated side of imadisj 5942. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑 → (dom 𝐴𝐵) ≠ ∅)       (𝜑 → (𝐴𝐵) ≠ ∅)
 
Theoremwnefimgd 40392 The image of a mapping from A is nonempty if A is nonempty. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ≠ ∅)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐴) ≠ ∅)
 
Theoremfco2d 40393 Natural deduction form of fco2 6527. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐺:𝐴𝐵)    &   (𝜑 → (𝐹𝐵):𝐵𝐶)       (𝜑 → (𝐹𝐺):𝐴𝐶)
 
Theoremwfximgfd 40394 The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐶𝐴)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐶) ∈ (𝐹𝐴))
 
Theoremextoimad 40395* If |f(x)| <= C for all x then it applies to all x in the image of |f(x)| (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 𝐶)       (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥𝐶)
 
Theoremimo72b2lem0 40396* Lemma for imo72b2 40406. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐺:ℝ⟶ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴𝐵))) = (2 · ((𝐹𝐴) · (𝐺𝐵))))    &   (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)       (𝜑 → ((abs‘(𝐹𝐴)) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
 
Theoremsuprleubrd 40397* Natural deduction form of specialized suprleub 11596. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∀𝑧𝐴 𝑧𝐵)       (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐵)
 
Theoremimo72b2lem2 40398* Lemma for imo72b2 40406. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → ∀𝑧 ∈ ℝ (abs‘(𝐹𝑧)) ≤ 𝐶)       (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶)
 
Theoremsyldbl2 40399 Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)
((𝜑𝜓) → (𝜓𝜃))       ((𝜑𝜓) → 𝜃)
 
Theoremfunfvima2d 40400 A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
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