P. 417 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

frege123 41601*

Lemma for frege124 41602. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))))

Theorem

frege124 41602

If 𝑋 is a result of an application of the single-valued procedure 𝑅 to 𝑌 and if 𝑀 follows 𝑌 in the 𝑅-sequence, then 𝑀 belongs to the 𝑅-sequence beginning with 𝑋. Proposition 124 of [Frege1879] p. 80. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)))

Theorem

frege125 41603

Lemma for frege126 41604. Proposition 125 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ ((𝑋((t+‘𝑅) ∪ I )𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)))))

Theorem

frege126 41604

If 𝑀 follows 𝑌 in the 𝑅-sequence and if the procedure 𝑅 is single-valued, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑀 or precedes 𝑀 in the 𝑅-sequence. Proposition 126 of [Frege1879] p. 81. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))))

Theorem

frege127 41605

Communte antecedents of frege126 41604. Proposition 127 of [Frege1879] p. 82. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌(t+‘𝑅)𝑀 → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))))

Theorem

frege128 41606

Lemma for frege129 41607. Proposition 128 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ ((𝑀((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))) → (Fun ^◡◡𝑅 → ((¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌) → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋)))))

Theorem

frege129 41607

If the procedure 𝑅 is single-valued and 𝑌 belongs to the 𝑅 -sequence begining with 𝑀 or precedes 𝑀 in the 𝑅-sequence, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence begining with 𝑀 or precedes 𝑀 in the 𝑅-sequence. Proposition 129 of [Frege1879] p. 83. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → ((¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌) → (𝑌𝑅𝑋 → (¬ 𝑋(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑋))))

Theorem

frege130 41608*

Lemma for frege131 41609. Proposition 130 of [Frege1879] p. 84. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑀 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ ((∀𝑏((¬ 𝑏(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑏) → ∀𝑎(𝑏𝑅𝑎 → (¬ 𝑎(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑎))) → 𝑅 hereditary ((^◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))) → (Fun ^◡◡𝑅 → 𝑅 hereditary ((^◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀}))))

Theorem

frege131 41609

If the procedure 𝑅 is single-valued, then the property of belonging to the 𝑅-sequence begining with 𝑀 or preceeding 𝑀 in the 𝑅-sequence is hereditary in the 𝑅-sequence. Proposition 131 of [Frege1879] p. 85. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑀 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ (Fun ^◡◡𝑅 → 𝑅 hereditary ((^◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})))

Theorem

frege132 41610

Lemma for frege133 41611. Proposition 132 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑀 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ ((𝑅 hereditary ((^◡(t+‘𝑅) “ {𝑀}) ∪ (((t+‘𝑅) ∪ I ) “ {𝑀})) → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌)))) → (Fun ^◡◡𝑅 → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌)))))

Theorem

frege133 41611

If the procedure 𝑅 is single-valued and if 𝑀 and 𝑌 follow 𝑋 in the 𝑅-sequence, then 𝑌 belongs to the 𝑅-sequence beginning with 𝑀 or precedes 𝑀 in the 𝑅-sequence. Proposition 133 of [Frege1879] p. 86. (Contributed by RP, 9-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑀 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑋(t+‘𝑅)𝑀 → (𝑋(t+‘𝑅)𝑌 → (¬ 𝑌(t+‘𝑅)𝑀 → 𝑀((t+‘𝑅) ∪ I )𝑌))))

20.31.4  Exploring Topology via Seifert and Threlfall

See Seifert and Threlfall: A Textbook Of Topology (1980) which is an English translation of Lehrbuch der Topologie (1934).

20.31.4.1  Equinumerosity of sets of relations and maps

Because ((2_o ↑_m 𝐵) ↑_m 𝐴) ≈ (2_o ↑_m (𝐴 × 𝐵)) ≈ ((2_o ↑_m 𝐴) ↑_m 𝐵) is an instance of the law of exponents: ((𝐶 ↑_m 𝐵) ↑_m 𝐴) ≈ (𝐶 ↑_m (𝐴 × 𝐵)) ≈ ((𝐶 ↑_m 𝐴) ↑_m 𝐵) we are led to see that (𝒫 𝐵 ↑_m 𝐴) ≈ 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴 ↑_m 𝐵) is true for any two sets, 𝐴 and 𝐵, and thus there exist one-to-one onto relations between each of these three sets of relations.

Theorem

enrelmap 41612

The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. See rfovf1od 41621 for a demonstration of a natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑_m 𝐴))

Theorem

enrelmapr 41613

The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. (Contributed by RP, 27-Apr-2021.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴 ↑_m 𝐵))

Theorem

enmappw 41614

The set of all mappings from one set to the powerset of the other is equinumerous to the set of all mappings from the second set to the powerset of the first. (Contributed by RP, 27-Apr-2021.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑_m 𝐴) ≈ (𝒫 𝐴 ↑_m 𝐵))

Theorem

enmappwid 41615

The set of all mappings from the powerset to the powerset is equinumerous to the set of all mappings from the set to the powerset of the powerset. (Contributed by RP, 27-Apr-2021.)

⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑_m 𝒫 𝐴) ≈ (𝒫 𝒫 𝐴 ↑_m 𝐴))

Theorem

rfovd 41616*

Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))

Theorem

rfovfvd 41617*

Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, and relation 𝑅. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝑅) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑅𝑦}))

Theorem

rfovfvfvd 41618*

Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, relation 𝑅, and left element 𝑋. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝒫 (𝐴 × 𝐵))    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ 𝐺 = (𝐹‘𝑅)    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) = {𝑦 ∈ 𝐵 ∣ 𝑋𝑅𝑦})

Theorem

rfovcnvf1od 41619*

Properties of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝐴) ∧ ^◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))

Theorem

rfovcnvd 41620*

Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → ^◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))

Theorem

rfovf1od 41621*

The value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, is a bijection. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝐴))

Theorem

rfovcnvfvd 41622*

Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, evaluated at function 𝐺. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐹 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ (𝒫 𝐵 ↑_m 𝐴))    ⇒   ⊢ (𝜑 → (^◡𝐹‘𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐺‘𝑥))})

Theorem

fsovd 41623*

Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))

Theorem

fsovrfovd 41624*

The operator which gives a 1-to-1 a mapping to a subset and a reverse mapping from elements can be composed from the operator which gives a 1-to-1 mapping between relations and functions to subsets and the converse operator. (Contributed by RP, 15-May-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑢 ∈ 𝑎 ↦ {𝑣 ∈ 𝑏 ∣ 𝑢𝑟𝑣})))    &   ⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ ^◡𝑠))    ⇒   ⊢ (𝜑 → (𝐴𝑂𝐵) = ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ ^◡(𝐴𝑅𝐵))))

Theorem

fsovfvd 41625*

Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, when applied to function 𝐹. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝐴))    ⇒   ⊢ (𝜑 → (𝐺‘𝐹) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

Theorem

fsovfvfvd 41626*

Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, when applied to function 𝐹 and element 𝑌. (Contributed by RP, 25-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝐴))    &   ⊢ 𝐻 = (𝐺‘𝐹)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻‘𝑌) = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})

Theorem

fsovfd 41627*

The operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, gives a function between two sets of functions. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑_m 𝐴)⟶(𝒫 𝐴 ↑_m 𝐵))

Theorem

fsovcnvlem 41628*

The 𝑂 operator, which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ 𝐻 = (𝐵𝑂𝐴)    ⇒   ⊢ (𝜑 → (𝐻 ∘ 𝐺) = ( I ↾ (𝒫 𝐵 ↑_m 𝐴)))

Theorem

fsovcnvd 41629*

The value of the converse (𝐴𝑂𝐵) is (𝐵𝑂𝐴), where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ 𝐻 = (𝐵𝑂𝐴)    ⇒   ⊢ (𝜑 → ^◡𝐺 = 𝐻)

Theorem

fsovcnvfvd 41630*

The value of the converse of (𝐴𝑂𝐵), where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, evaluated at function 𝐹. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐴 ↑_m 𝐵))    ⇒   ⊢ (𝜑 → (^◡𝐺‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)}))

Theorem

fsovf1od 41631*

The value of (𝐴𝑂𝐵) is a bijection, where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets. (Contributed by RP, 27-Apr-2021.)

⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝐺 = (𝐴𝑂𝐵)    ⇒   ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑_m 𝐴)–1-1-onto→(𝒫 𝐴 ↑_m 𝐵))

Theorem

dssmapfvd 41632*

Value of the duality operator for self-mappings of subsets of a base set, 𝐵. (Contributed by RP, 19-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))

Theorem

dssmapfv2d 41633*

Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹. (Contributed by RP, 19-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))    &   ⊢ 𝐺 = (𝐷‘𝐹)    ⇒   ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))

Theorem

dssmapfv3d 41634*

Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹 and subset 𝑆. (Contributed by RP, 19-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))    &   ⊢ 𝐺 = (𝐷‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    &   ⊢ 𝑇 = (𝐺‘𝑆)    ⇒   ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))

Theorem

dssmapnvod 41635*

For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is its own inverse, an involution. (Contributed by RP, 20-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ^◡𝐷 = 𝐷)

Theorem

dssmapf1od 41636*

For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is one-to-one and onto. (Contributed by RP, 21-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑_m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑_m 𝒫 𝐵))

Theorem

dssmap2d 41637*

For any base set 𝐵 the duality operator for self-mappings of subsets of that base set when composed with itself is the restricted identity operator. (Contributed by RP, 21-Apr-2021.)

⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑_m 𝒫 𝐵)))

20.31.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods

For any base set, 𝐵, an arbitrary mapping of subsets to subsets can be called a pseudoclosure (pseudointerior) function, 𝐾, with its dual of a pseudointerior (pseudoclosure), 𝐼, related by the involution in dssmapfvd 41632. As 𝐾 gains properties of the closure (interior) function of a topology on 𝐵, so does its dual gain corresponding properties of the interior (closure) function of that topology.

As (𝒫 𝐵 ↑_m 𝒫 𝐵) ≈ (𝒫 𝒫 𝐵 ↑_m 𝐵) there is also a natural isomorphism which maps from 𝐼 to 𝑁 (and likewise for 𝐾 and 𝑀, introduced below) which identically gains the properties of the neighborhood function of a topology (modified and restricted to operate on single points). A function dual to 𝑁, which Stadler and Stadler refer to as a convergent function, is represented by 𝑀 in this section.

Based on this and the early treatment of topology in Seifert and Threlfall, it seems reasonable to define a pseudotopology as defined in terms of its base set and one of these functions with theorems treating the equivalence of the other definitions and adding topological structure if enough properties hold true.

	Neighborhoods		Interior		Closure		Convergents	Theorems
Functions	𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)		𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)		𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)		𝑀 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)
Correspondences (assuming (𝑋 ∈ 𝐵 ∧ 𝑆 ∈ 𝒫 𝐵))	𝑆 ∈ (𝑁‘𝑋)	↔	𝑋 ∈ (𝐼‘𝑆)	↔	¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))	↔	¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑋)	ntrclselnel1 41674, ntrneiel 41698, neicvgel1 41736
	¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)	↔	¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆))	↔	𝑋 ∈ (𝐾‘𝑆)	↔	𝑆 ∈ (𝑀‘𝑋)
Neighborhoods	(𝑁‘𝑋)	=	{𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑠)}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑠))}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)}	ntrneifv3 41699, clsneifv3 41727, neicvgfv 41738
Interior	{𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)}	=	(𝐼‘𝑆)	=	(𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))	=	{𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑥)}	ntrneifv4 41702, ntrclsfv 41676, clsneifv4 41728
Closure	{𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}	=	(𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆)))	=	(𝐾‘𝑆)	=	{𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑀‘𝑥)}	clsneifv4 41728, ntrclsfv 41676, ntrneifv4 41702
Convergents	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑁‘𝑋)}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠))}	=	{𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐾‘𝑠)}	=	(𝑀‘𝑋)	neicvgfv 41738, clsneifv3 41727, ntrneifv3 41699

We have the following table of equivalences to axioms largely established by Kuratowski. In the formulas in this table, to reduce the width of the columns, if any of the variables 𝑥, 𝑠, or 𝑡 are used, then they are implicitly universally quantified and 𝑥 (respectively 𝑠 and 𝑡) ranges over 𝐵 (respectively 𝒫 𝐵 and 𝒫 𝐵).

Assuming a prefix of: ∀𝑥 ∈ 𝐵∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵	Neighborhoods	Interior	Closure	Convergents	Equivalence Theorems
K0' Neighborhoods are nonempty.	(𝑁‘𝑥) ≠ ∅	∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐼‘𝑢)	∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐾‘𝑢)	(𝑀‘𝑥) ≠ 𝒫 𝐵	ntrclsneine0 41682, ntrneineine0 41704, ntrneineine1 41705
KA' No neighborhood is equal to the full powerset.	(𝑁‘𝑥) ≠ 𝒫 𝐵	∃𝑢 ∈ 𝒫 𝐵¬ 𝑥 ∈ (𝐼‘𝑢)	∃𝑢 ∈ 𝒫 𝐵𝑥 ∈ (𝐾‘𝑢)	(𝑀‘𝑥) ≠ ∅	ntrclsneine0 41682, ntrneineine0 41704, ntrneineine1 41705
K0 Preservation of the Nullary Union of Closures	𝐵 ∈ (𝑁‘𝑥)	(𝐼‘𝐵) = 𝐵	(𝐾‘∅) = ∅	¬ ∅ ∈ (𝑀‘𝑥)	ntrclscls00 41683, ntrneicls00 41706, ntrneicls11 41707
KA Preservation of the Nullary Union of Interiors	¬ ∅ ∈ (𝑁‘𝑥)	(𝐼‘∅) = ∅	(𝐾‘𝐵) = 𝐵	𝐵 ∈ (𝑀‘𝑥)	ntrclscls00 41683, ntrneicls00 41706, ntrneicls11 41707
K1 Isotonic Montonic	((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑁‘𝑥))	(𝑠 ⊆ 𝑡 → (𝐼‘𝑠) ⊆ (𝐼‘𝑡)) — or — ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∪ 𝑡)) — or — (𝐼‘(𝑠 ∩ 𝑡)) ⊆ ((𝐼‘𝑠) ∩ (𝐼‘𝑡))	(𝑠 ⊆ 𝑡 → (𝐾‘𝑠) ⊆ (𝐾‘𝑡)) — or — ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) ⊆ (𝐾‘(𝑠 ∪ 𝑡)) — or — (𝐾‘(𝑠 ∩ 𝑡)) ⊆ ((𝐾‘𝑠) ∩ (𝐾‘𝑡))	((𝑠 ∈ (𝑀‘𝑥) ∧ 𝑠 ⊆ 𝑡) → 𝑡 ∈ (𝑀‘𝑥))	isotone1 41665, isotone2 41666, ntrclsiso 41684, ntrneiiso 41708
K2 Closure is Expansive	(𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠)	(𝐼‘𝑠) ⊆ 𝑠	𝑠 ⊆ (𝐾‘𝑠)	(𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑀‘𝑥))	ntrclsk2 41685, ntrneik2 41709, ntrneix2 41710
KB Non-disjoint Neighborhoods	((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ≠ ∅)	((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)	((𝑠 ∪ 𝑡) = 𝐵 → ((𝐾‘𝑠) ∪ (𝐾‘𝑡)) = 𝐵)	((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥)))	ntrclskb 41686, ntrneikb 41711, ntrneixb 41712
K3 Closure is Sub-linear	((𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)) → (𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥))	((𝐼‘𝑠) ∩ (𝐼‘𝑡)) ⊆ (𝐼‘(𝑠 ∩ 𝑡))	(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))	((𝑠 ∪ 𝑡) ∈ (𝑀‘𝑥) → (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥)))	ntrclsk3 41687, ntrneik3 41713, ntrneix3 41714
K13 Closure is finitely linear	((𝑠 ∩ 𝑡) ∈ (𝑁‘𝑥) ↔ (𝑠 ∈ (𝑁‘𝑥) ∧ 𝑡 ∈ (𝑁‘𝑥)))	(𝐼‘(𝑠 ∩ 𝑡)) = ((𝐼‘𝑠) ∩ (𝐼‘𝑡))	(𝐾‘(𝑠 ∪ 𝑡)) = ((𝐾‘𝑠) ∪ (𝐾‘𝑡))	((𝑠 ∪ 𝑡) ∈ (𝑀‘𝑥) ↔ (𝑠 ∈ (𝑀‘𝑥) ∨ 𝑡 ∈ (𝑀‘𝑥)))	ntrclsk13 41688, ntrneik13 41715, ntrneix13 41716
K4 Closure is idempotent	(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))	(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠)	(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)	(𝑠 ∈ (𝑀‘𝑥) ↔ ∃𝑢 ∈ (𝑀‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑀‘𝑦)))	ntrclsk4 41689, ntrneik4 41718

Using these properties as axiomic constraints on the functions, certain collections of them give rise to named spaces.

Space	Foundational Axioms	Derived Axioms	Theorems
Csázár Generalized Neighborhood Space	K2	KA', KA, KB	ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653
Min Strong Generalized Neighborhood Space	K2, K3	KA', KA, KB	ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653
Gniłka Extended Topology	K0', K1	K0	neik0pk1imk0 41664
Brissaud Space	K0, K2	K0', KA', KA, KB	neik0imk0p 41653, ntrk2imkb 41654, ntrkbimka 41655
Neighborhood Space	K0', K1, K2	K0, KA', KA, KB	neik0pk1imk0 41664, ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653
Davey and Priestley Intersection Structure	K1, K4
Moore Closure Space	K1, K2, K4	KA', KA, KB	ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653
Convex Closure Space	K0', K1, K2, K4	K0, KA', KA, KB	neik0pk1imk0 41664, ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653
Smyth Neighborhood Space	K0', K13	K0, K1, K3	neik0pk1imk0 41664, ntrk1k3eqk13 41667
Čech Closure Space Pretopological Space	K0', K2, K13	K0, K1, KA', KA, KB, K3	neik0pk1imk0 41664, ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653, ntrk1k3eqk13 41667
Topological Space	K0', K2, K13, K4	K0, K1, KA', KA, KB, K3	neik0pk1imk0 41664, ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653, ntrk1k3eqk13 41667
Alexandroff Space	K0', K2, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 41664, ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653, ntrk1k3eqk13 41667, TBD
Alexandroff Topological Space	K0', K2, K4, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 41664, ntrk2imkb 41654, ntrkbimka 41655, neik0imk0p 41653, ntrk1k3eqk13 41667, TBD

Theorem

or3or 41638

Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)

⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))

Theorem

andi3or 41639

Distribute over triple disjunction. (Contributed by RP, 5-Jul-2021.)

⊢ ((𝜑 ∧ (𝜓 ∨ 𝜒 ∨ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜑 ∧ 𝜃)))

Theorem

uneqsn 41640

If a union of classes is equal to a singleton then at least one class is equal to the singleton while the other may be equal to the empty set. (Contributed by RP, 5-Jul-2021.)

⊢ ((𝐴 ∪ 𝐵) = {𝐶} ↔ ((𝐴 = {𝐶} ∧ 𝐵 = {𝐶}) ∨ (𝐴 = {𝐶} ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ 𝐵 = {𝐶})))

Theorem

df3o2 41641

Ordinal 3 is the unordered triple containing ordinals 0, 1, and 2. (Contributed by RP, 8-Jul-2021.)

⊢ 3_o = {∅, 1_o, 2_o}

Theorem

df3o3 41642

Ordinal 3, fully expanded. (Contributed by RP, 8-Jul-2021.)

⊢ 3_o = {∅, {∅}, {∅, {∅}}}

Theorem

brfvimex 41643

If a binary relation holds and the relation is the value of a function, then the argument to that function is a set. (Contributed by RP, 22-May-2021.)

⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐹‘𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)

Theorem

brovmptimex 41644*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V))

Theorem

brovmptimex1 41645*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)

Theorem

brovmptimex2 41646*

If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.)

⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷))    ⇒   ⊢ (𝜑 → 𝐷 ∈ V)

Theorem

brcoffn 41647

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 7-Jun-2021.)

⊢ (𝜑 → 𝐶 Fn 𝑌)    &   ⊢ (𝜑 → 𝐷:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐷(𝐷‘𝐴) ∧ (𝐷‘𝐴)𝐶𝐵))

Theorem

brcofffn 41648

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶 Fn 𝑍)    &   ⊢ (𝜑 → 𝐷:𝑌⟶𝑍)    &   ⊢ (𝜑 → 𝐸:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝐸(𝐸‘𝐴) ∧ (𝐸‘𝐴)𝐷(𝐷‘(𝐸‘𝐴)) ∧ (𝐷‘(𝐸‘𝐴))𝐶𝐵))

Theorem

brco2f1o 41649

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)    &   ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ 𝐷)𝐵)    ⇒   ⊢ (𝜑 → ((^◡𝐶‘𝐵)𝐶𝐵 ∧ 𝐴𝐷(^◡𝐶‘𝐵)))

Theorem

brco3f1o 41650

Conditions allowing the decomposition of a binary relation. (Contributed by RP, 8-Jun-2021.)

⊢ (𝜑 → 𝐶:𝑌–1-1-onto→𝑍)    &   ⊢ (𝜑 → 𝐷:𝑋–1-1-onto→𝑌)    &   ⊢ (𝜑 → 𝐸:𝑊–1-1-onto→𝑋)    &   ⊢ (𝜑 → 𝐴(𝐶 ∘ (𝐷 ∘ 𝐸))𝐵)    ⇒   ⊢ (𝜑 → ((^◡𝐶‘𝐵)𝐶𝐵 ∧ (^◡𝐷‘(^◡𝐶‘𝐵))𝐷(^◡𝐶‘𝐵) ∧ 𝐴𝐸(^◡𝐷‘(^◡𝐶‘𝐵))))

Theorem

ntrclsbex 41651

If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then the base set exists. (Contributed by RP, 21-May-2021.)

⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)

Theorem

ntrclsrcomplex 41652

The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 25-Jun-2021.)

⊢ 𝐷 = (𝑂‘𝐵)    &   ⊢ (𝜑 → 𝐼𝐷𝐾)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Theorem

neik0imk0p 41653

Kuratowski's K0 axiom implies K0'. Neighborhood version. Also a proof the dual KA axiom implies KA' when considering the convergents. (Contributed by RP, 28-Jun-2021.)

⊢ (∀𝑥 ∈ 𝐵 𝐵 ∈ (𝑁‘𝑥) → ∀𝑥 ∈ 𝐵 (𝑁‘𝑥) ≠ ∅)

Theorem

ntrk2imkb 41654*

If an interior function is contracting, the interiors of disjoint sets are disjoint. Kuratowski's K2 axiom implies KB. Interior version. (Contributed by RP, 9-Jun-2021.)

⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))

Theorem

ntrkbimka 41655*

If the interiors of disjoint sets are disjoint, then the interior of the empty set is the empty set. (Contributed by RP, 14-Jun-2021.)

⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) → (𝐼‘∅) = ∅)

Theorem

ntrk0kbimka 41656*

If the interiors of disjoint sets are disjoint and the interior of the base set is the base set, then the interior of the empty set is the empty set. Obsolete version of ntrkbimka 41655. (Contributed by RP, 12-Jun-2021.)

⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵)) → (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘∅) = ∅))

Theorem

clsk3nimkb 41657*

If the base set is not empty, axiom K3 does not imply KB. A concrete example with a pseudo-closure function of 𝑘 = (𝑥 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ 𝑥)) is given. (Contributed by RP, 16-Jun-2021.)

⊢ ¬ ∀𝑏∀𝑘 ∈ (𝒫 𝑏 ↑_m 𝒫 𝑏)(∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏(𝑘‘(𝑠 ∪ 𝑡)) ⊆ ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) → ∀𝑠 ∈ 𝒫 𝑏∀𝑡 ∈ 𝒫 𝑏((𝑠 ∪ 𝑡) = 𝑏 → ((𝑘‘𝑠) ∪ (𝑘‘𝑡)) = 𝑏))

Theorem

clsk1indlem0 41658

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ (𝐾‘∅) = ∅

Theorem

clsk1indlem2 41659*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o𝑠 ⊆ (𝐾‘𝑠)

Theorem

clsk1indlem3 41660*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K3 property of being sub-linear. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o∀𝑡 ∈ 𝒫 3_o(𝐾‘(𝑠 ∪ 𝑡)) ⊆ ((𝐾‘𝑠) ∪ (𝐾‘𝑡))

Theorem

clsk1indlem4 41661*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) has the K4 property of idempotence. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∀𝑠 ∈ 𝒫 3_o(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)

Theorem

clsk1indlem1 41662*

The ansatz closure function (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟)) does not have the K1 property of isotony. (Contributed by RP, 6-Jul-2021.)

⊢ 𝐾 = (𝑟 ∈ 𝒫 3_o ↦ if(𝑟 = {∅}, {∅, 1_o}, 𝑟))    ⇒   ⊢ ∃𝑠 ∈ 𝒫 3_o∃𝑡 ∈ 𝒫 3_o(𝑠 ⊆ 𝑡 ∧ ¬ (𝐾‘𝑠) ⊆ (𝐾‘𝑡))

Theorem

clsk1independent 41663*

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 41664*

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 41665*

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 41666*

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 41667*

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 41668*

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 41669*

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 41670*

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 41671*

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 41672*

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 41673*

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 41674*

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 41675*

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 41676*

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 41677*

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 41678*

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 41679*

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 41680*

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 41681*

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 41682*

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 41683*

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 41684*

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 41685*

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 41686*

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 41687*

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 41688*

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 41689*

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 41690*

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 41691*

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 41692*

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 41693*

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 41694*

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 41695*

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 41696*

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 41697*

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 41698*

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 41699*

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 41700*

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) ≠ ∅))