P. 404 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

frege85 40301*

Commuted form of frege77 40293. Proposition 85 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴)))

Theorem

frege86 40302*

Conclusion about element one past 𝑌 in the 𝑅-sequence. Proposition 86 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (((𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴))) → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴)))))

Theorem

frege87 40303*

If 𝑍 is a result of an application of the procedure 𝑅 to an object 𝑌 that follows 𝑋 in the 𝑅-sequence and if every result of an application of the procedure 𝑅 to 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then 𝑍 has property 𝐴. Proposition 87 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴))))

Theorem

frege88 40304*

Commuted form of frege87 40303. Proposition 88 of [Frege1879] p. 67. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → 𝑍 ∈ 𝐴))))

Theorem

frege89 40305*

One direction of dffrege76 40292. Proposition 89 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → 𝑌 ∈ 𝑓)) → 𝑋(t+‘𝑅)𝑌)

Theorem

frege90 40306*

Add antecedent to frege89 40305. Proposition 90 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ ((𝜑 → ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → 𝑌 ∈ 𝑓))) → (𝜑 → 𝑋(t+‘𝑅)𝑌))

Theorem

frege91 40307

Every result of an application of a procedure 𝑅 to an object 𝑋 follows that 𝑋 in the 𝑅-sequence. Proposition 91 of [Frege1879] p. 68. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (𝑋𝑅𝑌 → 𝑋(t+‘𝑅)𝑌)

Theorem

frege92 40308

Inference from frege91 40307. Proposition 92 of [Frege1879] p. 69. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (𝑋 = 𝑍 → (𝑋𝑅𝑌 → 𝑍(t+‘𝑅)𝑌))

Theorem

frege93 40309*

Necessary condition for two elements to be related by the transitive closure. Proposition 93 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (∀𝑓(∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝑓) → (𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓)) → 𝑋(t+‘𝑅)𝑌)

Theorem

frege94 40310*

Looking one past a pair related by transitive closure of a relation. Proposition 94 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑍 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ ((𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → ∀𝑓(∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → (𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓)))) → (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → 𝑋(t+‘𝑅)𝑍)))

Theorem

frege95 40311

Looking one past a pair related by transitive closure of a relation. Proposition 95 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝐴    ⇒   ⊢ (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → 𝑋(t+‘𝑅)𝑍))

Theorem

frege96 40312

Every result of an application of the procedure 𝑅 to an object that follows 𝑋 in the 𝑅-sequence follows 𝑋 in the 𝑅 -sequence. Proposition 96 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝐴    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌𝑅𝑍 → 𝑋(t+‘𝑅)𝑍))

Theorem

frege97 40313

The property of following 𝑋 in the 𝑅-sequence is hereditary in the 𝑅-sequence. Proposition 97 of [Frege1879] p. 71.

Here we introduce the image of a singleton under a relation as class which stands for the property of following 𝑋 in the 𝑅 -sequence. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋})

Theorem

frege98 40314

If 𝑌 follows 𝑋 and 𝑍 follows 𝑌 in the 𝑅-sequence then 𝑍 follows 𝑋 in the 𝑅-sequence because the transitive closure of a relation has the transitive property. Proposition 98 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 6-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑍 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍))

20.31.3.10  _Begriffsschrift_ Chapter III Member of sequence

𝑝((t+‘𝑅) ∪ I )𝑐 means 𝑐 is a member of the 𝑅 -sequence begining with 𝑝 and 𝑝 is a member of the 𝑅 -sequence ending with 𝑐.

dffrege99 40315 through frege114 40330 develop this.

This will be shown to be related to the transitive-reflexive closure of relation 𝑅. But more work needs to be done on transitive closure of relations before this is ready for Metamath.

Theorem

dffrege99 40315

If 𝑍 is identical with 𝑋 or follows 𝑋 in the 𝑅 -sequence, then we say : "𝑍 belongs to the 𝑅-sequence beginning with 𝑋 " or "𝑋 belongs to the 𝑅-sequence ending with 𝑍". Definition 99 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.)

⊢ 𝑍 ∈ 𝑈    ⇒   ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Theorem

frege100 40316

One direction of dffrege99 40315. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑈    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))

Theorem

frege101 40317

Lemma for frege102 40318. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑈    ⇒   ⊢ ((𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉))))

Theorem

frege102 40318

If 𝑍 belongs to the 𝑅-sequence beginning with 𝑋, then every result of an application of the procedure 𝑅 to 𝑍 follows 𝑋 in the 𝑅-sequence. Proposition 102 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝐴    &   ⊢ 𝑍 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉))

Theorem

frege103 40319

Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((𝑍 = 𝑋 → 𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)))

Theorem

frege104 40320

Proposition 104 of [Frege1879] p. 73.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍))

Theorem

frege105 40321

Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)

Theorem

frege106 40322

Whatever follows 𝑋 in the 𝑅-sequence belongs to the 𝑅 -sequence beginning with 𝑋. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍)

Theorem

frege107 40323

Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑉 ∈ 𝐴    ⇒   ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉)))

Theorem

frege108 40324

If 𝑌 belongs to the 𝑅-sequence beginning with 𝑍, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑍. Proposition 108 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → 𝑍((t+‘𝑅) ∪ I )𝑉))

Theorem

frege109 40325

The property of belonging to the 𝑅-sequence beginning with 𝑋 is hereditary in the 𝑅-sequence. Proposition 109 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑉    ⇒   ⊢ 𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋})

Theorem

frege110 40326*

Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑀 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))

Theorem

frege111 40327

If 𝑌 belongs to the 𝑅-sequence beginning with 𝑍, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑍 or precedes 𝑍 in the 𝑅-sequence. Proposition 111 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Revised by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → (¬ 𝑉(t+‘𝑅)𝑍 → 𝑍((t+‘𝑅) ∪ I )𝑉)))

Theorem

frege112 40328

Identity implies belonging to the 𝑅-sequence beginning with self. Proposition 112 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑍 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)

Theorem

frege113 40329

Proposition 113 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)))

Theorem

frege114 40330

If 𝑋 belongs to the 𝑅-sequence beginning with 𝑍, then 𝑍 belongs to the 𝑅-sequence beginning with 𝑋 or 𝑋 follows 𝑍 in the 𝑅-sequence. Proposition 114 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))

20.31.3.11  _Begriffsschrift_ Chapter III Single-valued procedures

Fun ^◡◡𝑅 means the relationship content of procedure 𝑅 is single-valued. The double converse allows us to simply apply this syntax in place of Frege's even though the original never explicitly limited discussion of propositional statements which vary on two variables to relations.

dffrege115 40331 through frege133 40349 develop this and how functions relate to transitive and transitive-reflexive closures.

Theorem

dffrege115 40331*

If from the circumstance that 𝑐 is a result of an application of the procedure 𝑅 to 𝑏, whatever 𝑏 may be, it can be inferred that every result of an application of the procedure 𝑅 to 𝑏 is the same as 𝑐, then we say : "The procedure 𝑅 is single-valued". Definition 115 of [Frege1879] p. 77. (Contributed by RP, 7-Jul-2020.)

⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ^◡◡𝑅)

Theorem

frege116 40332*

One direction of dffrege115 40331. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    ⇒   ⊢ (Fun ^◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))

Theorem

frege117 40333*

Lemma for frege118 40334. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    ⇒   ⊢ ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))))

Theorem

frege118 40334*

Simplified application of one direction of dffrege115 40331. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))

Theorem

frege119 40335*

Lemma for frege120 40336. Proposition 119 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))))

Theorem

frege120 40336

Simplified application of one direction of dffrege115 40331. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝐴 ∈ 𝑊    ⇒   ⊢ (Fun ^◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))

Theorem

frege121 40337

Lemma for frege122 40338. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

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

Theorem

frege122 40338

If 𝑋 is a result of an application of the single-valued procedure 𝑅 to 𝑌, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑋. Proposition 122 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

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

Theorem

frege123 40339*

Lemma for frege124 40340. 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 40340

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 40341

Lemma for frege126 40342. 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 40342

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 40343

Communte antecedents of frege126 40342. 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 40344

Lemma for frege129 40345. 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 40345

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 40346*

Lemma for frege131 40347. 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 40347

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 40348

Lemma for frege133 40349. 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 40349

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 40350

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 40359 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 40351

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 40352

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 40353

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 40354*

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 40355*

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 40356*

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 40357*

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 40358*

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 40359*

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 40360*

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 40361*

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 40362*

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 40363*

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 40364*

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 40365*

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 40366*

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 40367*

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 40368*

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 40369*

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 40370*

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 40371*

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 40372*

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 40373*

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 40374*

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 40375*

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 40370. 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 40414, ntrneiel 40438, neicvgel1 40476
	¬ (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 40439, clsneifv3 40467, neicvgfv 40478
Interior	{𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)}	=	(𝐼‘𝑆)	=	(𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))	=	{𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑥)}	ntrneifv4 40442, ntrclsfv 40416, clsneifv4 40468
Closure	{𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)}	=	(𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆)))	=	(𝐾‘𝑆)	=	{𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑀‘𝑥)}	clsneifv4 40468, ntrclsfv 40416, ntrneifv4 40442
Convergents	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑁‘𝑋)}	=	{𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠))}	=	{𝑠 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐾‘𝑠)}	=	(𝑀‘𝑋)	neicvgfv 40478, clsneifv3 40467, ntrneifv3 40439

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 40422, ntrneineine0 40444, ntrneineine1 40445
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 40422, ntrneineine0 40444, ntrneineine1 40445
K0 Preservation of the Nullary Union of Closures	𝐵 ∈ (𝑁‘𝑥)	(𝐼‘𝐵) = 𝐵	(𝐾‘∅) = ∅	¬ ∅ ∈ (𝑀‘𝑥)	ntrclscls00 40423, ntrneicls00 40446, ntrneicls11 40447
KA Preservation of the Nullary Union of Interiors	¬ ∅ ∈ (𝑁‘𝑥)	(𝐼‘∅) = ∅	(𝐾‘𝐵) = 𝐵	𝐵 ∈ (𝑀‘𝑥)	ntrclscls00 40423, ntrneicls00 40446, ntrneicls11 40447
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 40405, isotone2 40406, ntrclsiso 40424, ntrneiiso 40448
K2 Closure is Expansive	(𝑠 ∈ (𝑁‘𝑥) → 𝑥 ∈ 𝑠)	(𝐼‘𝑠) ⊆ 𝑠	𝑠 ⊆ (𝐾‘𝑠)	(𝑥 ∈ 𝑠 → 𝑠 ∈ (𝑀‘𝑥))	ntrclsk2 40425, ntrneik2 40449, ntrneix2 40450
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 40426, ntrneikb 40451, ntrneixb 40452
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 40427, ntrneik3 40453, ntrneix3 40454
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 40428, ntrneik13 40455, ntrneix13 40456
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 40429, ntrneik4 40458

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 40394, ntrkbimka 40395, neik0imk0p 40393
Min Strong Generalized Neighborhood Space	K2, K3	KA', KA, KB	ntrk2imkb 40394, ntrkbimka 40395, neik0imk0p 40393
Gniłka Extended Topology	K0', K1	K0	neik0pk1imk0 40404
Brissaud Space	K0, K2	K0', KA', KA, KB	neik0imk0p 40393, ntrk2imkb 40394, ntrkbimka 40395
Neighborhood Space	K0', K1, K2	K0, KA', KA, KB	neik0pk1imk0 40404, ntrk2imkb 40394, ntrkbimka 40395, neik0imk0p 40393
Davey and Priestley Intersection Structure	K1, K4
Moore Closure Space	K1, K2, K4	KA', KA, KB	ntrk2imkb 40394, ntrkbimka 40395, neik0imk0p 40393
Convex Closure Space	K0', K1, K2, K4	K0, KA', KA, KB	neik0pk1imk0 40404, ntrk2imkb 40394, ntrkbimka 40395, neik0imk0p 40393
Smyth Neighborhood Space	K0', K13	K0, K1, K3	neik0pk1imk0 40404, ntrk1k3eqk13 40407
Čech Closure Space Pretopological Space	K0', K2, K13	K0, K1, KA', KA, KB, K3	neik0pk1imk0 40404, ntrk2imkb 40394, ntrkbimka 40395, neik0imk0p 40393, ntrk1k3eqk13 40407
Topological Space	K0', K2, K13, K4	K0, K1, KA', KA, KB, K3	neik0pk1imk0 40404, ntrk2imkb 40394, ntrkbimka 40395, neik0imk0p 40393, ntrk1k3eqk13 40407
Alexandroff Space	K0', K2, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 40404, ntrk2imkb 40394, ntrkbimka 40395, neik0imk0p 40393, ntrk1k3eqk13 40407, TBD
Alexandroff Topological Space	K0', K2, K4, K5	K0, K1, KA', KA, KB, K3, K13	neik0pk1imk0 40404, ntrk2imkb 40394, ntrkbimka 40395, neik0imk0p 40393, ntrk1k3eqk13 40407, TBD

Theorem

sscon34b 40376

Relative complementation reverses inclusion of subclasses. Relativized version of complss 4125. (Contributed by RP, 3-Jun-2021.)

⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)))

Theorem

rcompleq 40377

Two subclasses are equal if and only if their relative complements are equal. Relativized version of compleq 4126. (Contributed by RP, 10-Jun-2021.)

⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 = 𝐵 ↔ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)))

Theorem

or3or 40378

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

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

Theorem

andi3or 40379

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

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

Theorem

uneqsn 40380

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 40381

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

⊢ 3_o = {∅, 1_o, 2_o}

Theorem

df3o3 40382

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

⊢ 3_o = {∅, {∅}, {∅, {∅}}}

Theorem

brfvimex 40383

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 40384*

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 40385*

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 40386*

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 40387

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 40388

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 40389

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 40390

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 40391

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 40392

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 40393

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 40394*

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 40395*

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 40396*

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 40395. (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 40397*

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 40398

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 40399*

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 40400*

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))