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Theorem List for Intuitionistic Logic Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisstruct2r 11501 The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘𝑋))) → 𝐹 Struct 𝑋)

Theoremstructex 11502 A structure is a set. (Contributed by AV, 10-Nov-2021.)
(𝐺 Struct 𝑋𝐺 ∈ V)

Theoremstructn0fun 11503 A structure without the empty set is a function. (Contributed by AV, 13-Nov-2021.)
(𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}))

Theoremisstructim 11504 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(𝐹 Struct ⟨𝑀, 𝑁⟩ → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))

Theoremisstructr 11505 The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
(((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)

Theoremstructcnvcnv 11506 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
(𝐹 Struct 𝑋𝐹 = (𝐹 ∖ {∅}))

Theoremstructfung 11507 The converse of the converse of a structure is a function. Closed form of structfun 11508. (Contributed by AV, 12-Nov-2021.)
(𝐹 Struct 𝑋 → Fun 𝐹)

Theoremstructfun 11508 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.)
𝐹 Struct 𝑋       Fun 𝐹

Theoremstructfn 11509 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐹 Struct ⟨𝑀, 𝑁       (Fun 𝐹 ∧ dom 𝐹 ⊆ (1...𝑁))

Theoremslotfni 11510 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       𝐸 Fn V

Theoremstrnfvnd 11511 Deduction version of strnfvn 11513. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (𝜑𝑆𝑉)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐸𝑆) = (𝑆𝑁))

Theorembasfn 11512 The base set extractor is a function on V. (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V

Theoremstrnfvn 11513 Value of a structure component extractor 𝐸. Normally, 𝐸 is a defined constant symbol such as Base (df-base 11496) and 𝑁 is a fixed integer such as 1. 𝑆 is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strnfv 11533. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

𝑆 ∈ V    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸𝑆) = (𝑆𝑁)

Theoremstrfvssn 11514 A structure component extractor produces a value which is contained in a set dependent on 𝑆, but not 𝐸. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 19-Jan-2023.)
𝐸 = Slot 𝑁    &   (𝜑𝑆𝑉)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐸𝑆) ⊆ ran 𝑆)

Theoremndxarg 11515 Get the numeric argument from a defined structure component extractor such as df-base 11496. (Contributed by Mario Carneiro, 6-Oct-2013.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝐸‘ndx) = 𝑁

Theoremndxid 11516 A structure component extractor is defined by its own index. This theorem, together with strnfv 11533 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the 1 in df-base 11496, making it easier to change should the need arise.

(Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.)

𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       𝐸 = Slot (𝐸‘ndx)

Theoremstrndxid 11517 The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
(𝜑𝑆𝑉)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ       (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸𝑆))

Theoremreldmsets 11518 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Rel dom sSet

Theoremsetsvalg 11519 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝑆𝑉𝐴𝑊) → (𝑆 sSet 𝐴) = ((𝑆 ↾ (V ∖ dom {𝐴})) ∪ {𝐴}))

Theoremsetsvala 11520 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.)
((𝑆𝑉𝐴𝑋𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))

Theoremsetsex 11521 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
((𝑆𝑉𝐴𝑋𝐵𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V)

Theoremsetsssvald 11522 Value of the structure replacement function, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 21-Jan-2023.)
𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → (𝐸‘ndx) ∈ dom 𝑆)       (𝜑 → (𝑆 sSet ⟨(𝐸‘ndx), (𝐸𝑆)⟩) ⊆ 𝑆)

Theoremfvsetsid 11523 The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.)
((𝐹𝑉𝑋𝑊𝑌𝑈) → ((𝐹 sSet ⟨𝑋, 𝑌⟩)‘𝑋) = 𝑌)

Theoremsetsfun 11524 A structure with replacement is a function if the original structure is a function. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun 𝐺) ∧ (𝐼𝑈𝐸𝑊)) → Fun (𝐺 sSet ⟨𝐼, 𝐸⟩))

Theoremsetsfun0 11525 A structure with replacement without the empty set is a function if the original structure without the empty set is a function. This variant of setsfun 11524 is useful for proofs based on isstruct2r 11501 which requires Fun (𝐹 ∖ {∅}) for 𝐹 to be an extensible structure. (Contributed by AV, 7-Jun-2021.)
(((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ (𝐼𝑈𝐸𝑊)) → Fun ((𝐺 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Theoremsetsn0fun 11526 The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set)is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
(𝜑𝑆 Struct 𝑋)    &   (𝜑𝐼𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅}))

Theoremsetsresg 11527 The structure replacement function does not affect the value of 𝑆 away from 𝐴. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
((𝑆𝑉𝐴𝑊𝐵𝑋) → ((𝑆 sSet ⟨𝐴, 𝐵⟩) ↾ (V ∖ {𝐴})) = (𝑆 ↾ (V ∖ {𝐴})))

Theoremsetsabsd 11528 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
(𝜑𝑆𝑉)    &   (𝜑𝐴𝑊)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑈)       (𝜑 → ((𝑆 sSet ⟨𝐴, 𝐵⟩) sSet ⟨𝐴, 𝐶⟩) = (𝑆 sSet ⟨𝐴, 𝐶⟩))

Theoremsetscom 11529 Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝑆𝑉𝐴𝐵) ∧ (𝐶𝑊𝐷𝑋)) → ((𝑆 sSet ⟨𝐴, 𝐶⟩) sSet ⟨𝐵, 𝐷⟩) = ((𝑆 sSet ⟨𝐵, 𝐷⟩) sSet ⟨𝐴, 𝐶⟩))

Theoremstrnfvd 11530 Deduction version of strnfv 11533. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑 → (𝐸‘ndx) ∈ ℕ)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑𝐶 = (𝐸𝑆))

Theoremstrnfv2d 11531 Deduction version of strnfv 11533. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 23-Jan-2023.)
𝐸 = Slot (𝐸‘ndx)    &   (𝜑 → (𝐸‘ndx) ∈ ℕ)    &   (𝜑𝑆𝑉)    &   (𝜑 → Fun 𝑆)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)    &   (𝜑𝐶𝑊)       (𝜑𝐶 = (𝐸𝑆))

Theoremstrnfv2 11532 A variation on strnfv 11533 to avoid asserting that 𝑆 itself is a function, which involves sethood of all the ordered pair components of 𝑆. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 23-Jan-2023.)
𝑆 ∈ V    &   Fun 𝑆    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐶𝑉𝐶 = (𝐸𝑆))

Theoremstrnfv 11533 Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 11496). By virtue of ndxid 11516, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 23-Jan-2023.)
𝑆 Struct 𝑋    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆       (𝐶𝑉𝐶 = (𝐸𝑆))

Theoremstrnfv3 11534 Variant on strnfv 11533 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 23-Jan-2023.)
(𝜑𝑈 = 𝑆)    &   𝑆 Struct 𝑋    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆    &   (𝜑𝐶𝑉)    &   𝐴 = (𝐸𝑈)       (𝜑𝐴 = 𝐶)

Theoremstrnssd 11535 Deduction version of strnss 11536. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 23-Jan-2023.)
𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   (𝜑𝑇𝑉)    &   (𝜑 → Fun 𝑇)    &   (𝜑𝑆𝑇)    &   (𝜑 → ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆)       (𝜑 → (𝐸𝑇) = (𝐸𝑆))

Theoremstrnss 11536 Propagate component extraction to a structure 𝑇 from a subset structure 𝑆. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 23-Jan-2023.)
𝑇 ∈ V    &   Fun 𝑇    &   𝑆𝑇    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   ⟨(𝐸‘ndx), 𝐶⟩ ∈ 𝑆       (𝐸𝑇) = (𝐸𝑆)

Theoremstrn0 11537 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 23-Jan-2023.)
𝐹 = Slot 𝐼    &   𝐼 ∈ ℕ       ∅ = (𝐹‘∅)

Theorembase0 11538 The base set of the empty structure. (Contributed by David A. Wheeler, 7-Jul-2016.)
∅ = (Base‘∅)

Theoremsetsidn 11539 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ       ((𝑊𝐴𝐶𝑉) → 𝐶 = (𝐸‘(𝑊 sSet ⟨(𝐸‘ndx), 𝐶⟩)))

Theoremsetsnidn 11540 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ∈ ℕ    &   (𝐸‘ndx) ≠ 𝐷    &   𝐷 ∈ ℕ       ((𝑊𝐴𝐶𝑉) → (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩)))

Theorembaseval 11541 Value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
𝐾 ∈ V       (Base‘𝐾) = (𝐾‘1)

Theorembaseid 11542 Utility theorem: index-independent form of df-base 11496. (Contributed by NM, 20-Oct-2012.)
Base = Slot (Base‘ndx)

Theorembasendx 11543 Index value of the base set extractor. (Normally it is preferred to work with (Base‘ndx) rather than the hard-coded 1 in order to make structure theorems portable. This is an example of how to obtain it when needed.) (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.)
(Base‘ndx) = 1

Theorembasendxnn 11544 The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.)
(Base‘ndx) ∈ ℕ

Theoremreldmress 11545 The structure restriction is a proper operator, so it can be used with ovprc1 5685. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Rel dom ↾s

Theoremressid2 11546 General behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 26-Jan-2023.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)

Theoremressval2 11547 Value of nontrivial structure restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝑅 = (𝑊s 𝐴)    &   𝐵 = (Base‘𝑊)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))

Theoremressid 11548 Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
𝐵 = (Base‘𝑊)       (𝑊𝑋 → (𝑊s 𝐵) = 𝑊)

PART 6  BASIC TOPOLOGY

6.1  Topology

6.1.1  Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.

6.1.1.1  Topologies

Syntaxctop 11549 Syntax for the class of topologies.
class Top

Definitiondf-top 11550* Define the class of topologies. It is a proper class. See istopg 11551 and istopfin 11552 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}

Theoremistopg 11551* Express the predicate "𝐽 is a topology". See istopfin 11552 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))

Theoremistopfin 11552* Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 11551. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
(𝐽 ∈ Top → (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥((𝑥𝐽𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐽)))

Theoremuniopn 11553 The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)

Theoremiunopn 11554* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)

Theoreminopn 11555 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theoremfiinopn 11556 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top → ((𝐴𝐽𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴𝐽))

Theoremunopn 11557 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theorem0opn 11558 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top → ∅ ∈ 𝐽)

Theorem0ntop 11559 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
¬ ∅ ∈ Top

Theoremtopopn 11560 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋𝐽)

Theoremeltopss 11561 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)

6.1.1.2  Topologies on sets

Syntaxctopon 11562 Syntax for the function of topologies on sets.
class TopOn

Definitiondf-topon 11563* Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})

Theoremfuntopon 11564 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn

Theoremistopon 11565 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))

Theoremtopontop 11566 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)

Theoremtoponuni 11567 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)

Theoremtopontopi 11568 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐽 ∈ Top

Theoremtoponunii 11569 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐵 = 𝐽

Theoremtoptopon 11570 Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))

Theoremtoptopon2 11571 A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))

Theoremtopontopon 11572 A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘ 𝐽))

Theoremtoponsspwpwg 11573 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
(𝐴𝑉 → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴)

Theoremdmtopon 11574 The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V

Theoremfntopon 11575 The class TopOn is a function with domain V. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V

Theoremtoponmax 11576 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵𝐽)

Theoremtoponss 11577 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)

Theoremtoponcom 11578 If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘ 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐾))

Theoremtoponcomb 11579 Biconditional form of toponcom 11578. (Contributed by BJ, 5-Dec-2021.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘ 𝐾) ↔ 𝐾 ∈ (TopOn‘ 𝐽)))

Theoremtopgele 11580 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
(𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))

6.2  Metric spaces

6.2.1  Topological definitions using the reals

Syntaxccncf 11581 Extend class notation to include the operation which returns a class of continuous complex functions.
class cn

Definitiondf-cncf 11582* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦𝑎 ((abs‘(𝑥𝑦)) < 𝑑 → (abs‘((𝑓𝑥) − (𝑓𝑦))) < 𝑒)})

Theoremcncfval 11583* The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})

Theoremelcncf 11584* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))

Theoremelcncf2 11585* Version of elcncf 11584 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝑥)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝑥))) < 𝑦))))

Theoremcncfrss 11586 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)

Theoremcncfrss2 11587 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)

Theoremcncff 11588 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)

Theoremcncfi 11589* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
((𝐹 ∈ (𝐴cn𝐵) ∧ 𝐶𝐴𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝐶)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝐶))) < 𝑅))

Theoremelcncf1di 11590* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑 → ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+))    &   (𝜑 → (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))       (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵)))

Theoremelcncf1ii 11591* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
𝐹:𝐴𝐵    &   ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)    &   (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵))

Theoremrescncf 11592 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))

Theoremcncffvrn 11593 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴cn𝐵)) → (𝐹 ∈ (𝐴cn𝐶) ↔ 𝐹:𝐴𝐶))

Theoremcncfss 11594 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐵𝐶𝐶 ⊆ ℂ) → (𝐴cn𝐵) ⊆ (𝐴cn𝐶))

Theoremclimcncf 11595 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺:𝑍𝐴)    &   (𝜑𝐺𝐷)    &   (𝜑𝐷𝐴)       (𝜑 → (𝐹𝐺) ⇝ (𝐹𝐷))

Theoremabscncf 11596 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)

Theoremrecncf 11597 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℜ ∈ (ℂ–cn→ℝ)

Theoremimcncf 11598 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℑ ∈ (ℂ–cn→ℝ)

Theoremcjcncf 11599 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
∗ ∈ (ℂ–cn→ℂ)

Theoremmulc1cncf 11600* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

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