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Theorem List for Metamath Proof Explorer - 32001-32100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembj-xtagex 32001 The product of a set and the tagging of a set is a set. (Contributed by BJ, 2-Apr-2019.)
(𝐴𝑉 → (𝐵𝑊 → (𝐴 × tag 𝐵) ∈ V))

20.14.5.13  Tuples of classes

This subsection gives a definition of an ordered pair, or couple (2-tuple), which "works" for proper classes, as evidenced by Theorems bj-2uplth 32033 and bj-2uplex 32034 (but more importantly, bj-pr21val 32025 and bj-pr22val 32031). In particular, one can define well-behaved tuples of classes. Note, however, that classes in ZF(C) are only virtual, and in particular they cannot be quantified over.

The projections are denoted by pr1 and pr2 and the couple with projections (or coordinates) 𝐴 and 𝐵 is denoted by 𝐴, 𝐵.

Note that this definition uses the Kuratowksi definition (df-op 4035) as a preliminary definition, and then "redefines" a couple. It could also use the "short" version of the Kuratowski pair (see opthreg 8274) without needing the axiom of regularity; it could even bypass this definition by "inlining" it.

This definition is due to Anthony Morse and is expounded (with idiosyncratic notation) in

Anthony P. Morse, A Theory of Sets, Academic Press, 1965 (second edition 1986).

Note that this extends in a natural way to tuples.

A variation of this definition is justified in opthprc 4983, but here we use "tagged versions" of the factors (see df-bj-tag 31987) so that an m-tuple can equal an n-tuple only when m = n (and the projections are the same).

A comparison of the different definitions of tuples (strangely not mentioning Morse's), is given in

Dominic McCarty and Dana Scott, Reconsidering ordered pairs, Bull. Symbolic Logic, Volume 14, Issue 3 (Sept. 2008), 379--397.

where a recursive definition of tuples is given that avoids the 2-step definition of tuples and that can be adapted to various set theories.

Finally, another survey is

Akihiro Kanamori, The empty set, the singleton, and the ordered pair, Bull. Symbolic Logic, Volume 9, Number 3 (Sept. 2003), 273--298. (available at http://math.bu.edu/people/aki/8.pdf)

Syntaxbj-cproj 32002 Syntax for the class projection. (Contributed by BJ, 6-Apr-2019.)
class (𝐴 Proj 𝐵)

Definitiondf-bj-proj 32003* Definition of the class projection corresponding to tagged tuples. The expression (𝐴 Proj 𝐵) denotes the projection on the A^th component. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
(𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}

Theorembj-projeq 32004 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))

Theorembj-projeq2 32005 Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
(𝐵 = 𝐶 → (𝐴 Proj 𝐵) = (𝐴 Proj 𝐶))

Theorembj-projun 32006 The class projection on a given component preserves unions. (Contributed by BJ, 6-Apr-2019.)
(𝐴 Proj (𝐵𝐶)) = ((𝐴 Proj 𝐵) ∪ (𝐴 Proj 𝐶))

Theorembj-projex 32007 Sethood of the class projection. (Contributed by BJ, 6-Apr-2019.)
(𝐵𝑉 → (𝐴 Proj 𝐵) ∈ V)

Theorembj-projval 32008 Value of the class projection (proof can be shortened by 19 bytes by using sylancl3 31572). (Contributed by BJ, 6-Apr-2019.)
(𝐴𝑉 → (𝐴 Proj ({𝐵} × tag 𝐶)) = if(𝐵 = 𝐴, 𝐶, ∅))

Syntaxbj-c1upl 32009 Syntax for Morse monuple. (Contributed by BJ, 6-Apr-2019.)
class 𝐴

Definitiondf-bj-1upl 32010 Definition of the Morse monuple (1-tuple). This is not useful per se, but is used as a step towards the definition of couples (2-tuples, or ordered pairs). The reason for "tagging" the set is so that an m-tuple and an n-tuple be equal only when m = n. Note that with this definition, the 0-tuple is the empty set. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 32024, bj-2uplth 32033, bj-2uplex 32034, and the properties of the projections (see df-bj-pr1 32013 and df-bj-pr2 32027). (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
𝐴⦆ = ({∅} × tag 𝐴)

Theorembj-1upleq 32011 Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)

Syntaxbj-cpr1 32012 Syntax for the first class tuple projection. (Contributed by BJ, 6-Apr-2019.)
class pr1 𝐴

Definitiondf-bj-pr1 32013 Definition of the first projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr1eq 32014, bj-pr11val 32017, bj-pr21val 32025, bj-pr1ex 32018. (Contributed by BJ, 6-Apr-2019.) (New usage is discouraged.)
pr1 𝐴 = (∅ Proj 𝐴)

Theorembj-pr1eq 32014 Substitution property for pr1. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = 𝐵 → pr1 𝐴 = pr1 𝐵)

Theorembj-pr1un 32015 The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Theorembj-pr1val 32016 Value of the first projection. (Contributed by BJ, 6-Apr-2019.)
pr1 ({𝐴} × tag 𝐵) = if(𝐴 = ∅, 𝐵, ∅)

Theorembj-pr11val 32017 Value of the first projection of a monuple. (Contributed by BJ, 6-Apr-2019.)
pr1𝐴⦆ = 𝐴

Theorembj-pr1ex 32018 Sethood of the first projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr1 𝐴 ∈ V)

Theorembj-1uplth 32019 The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019.)
(⦅𝐴⦆ = ⦅𝐵⦆ ↔ 𝐴 = 𝐵)

Theorembj-1uplex 32020 A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.)
(⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V)

Theorembj-1upln0 32021 A monuple is nonempty. (Contributed by BJ, 6-Apr-2019.)
𝐴⦆ ≠ ∅

Syntaxbj-c2uple 32022 Syntax for Morse couple. (Contributed by BJ, 6-Oct-2018.)
class 𝐴, 𝐵

Definitiondf-bj-2upl 32023 Definition of the Morse couple. See df-bj-1upl 32010. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-2upleq 32024, bj-2uplth 32033, bj-2uplex 32034, and the properties of the projections (see df-bj-pr1 32013 and df-bj-pr2 32027). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐵))

Theorembj-2upleq 32024 Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Theorembj-pr21val 32025 Value of the first projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr1𝐴, 𝐵⦆ = 𝐴

Syntaxbj-cpr2 32026 Syntax for the second class tuple projection. (Contributed by BJ, 6-Oct-2018.)
class pr2 𝐴

Definitiondf-bj-pr2 32027 Definition of the second projection of a class tuple. New usage is discouraged because the precise definition is generally unimportant compared to the characteristic properties bj-pr2eq 32028, bj-pr22val 32031, bj-pr2ex 32032. (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
pr2 𝐴 = (1𝑜 Proj 𝐴)

Theorembj-pr2eq 32028 Substitution property for pr2. (Contributed by BJ, 6-Oct-2018.)
(𝐴 = 𝐵 → pr2 𝐴 = pr2 𝐵)

Theorembj-pr2un 32029 The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Theorembj-pr2val 32030 Value of the second projection. (Contributed by BJ, 6-Apr-2019.)
pr2 ({𝐴} × tag 𝐵) = if(𝐴 = 1𝑜, 𝐵, ∅)

Theorembj-pr22val 32031 Value of the second projection of a couple. (Contributed by BJ, 6-Oct-2018.)
pr2𝐴, 𝐵⦆ = 𝐵

Theorembj-pr2ex 32032 Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
(𝐴𝑉 → pr2 𝐴 ∈ V)

Theorembj-2uplth 32033 The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 4769). (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theorembj-2uplex 32034 A couple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Oct-2018.)
(⦅𝐴, 𝐵⦆ ∈ V ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembj-2upln0 32035 A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ∅

Theorembj-2upln1upl 32036 A couple is never equal to a monuple. It is in order to have this "non-clashing" result that tagging was used. Without tagging, we would have 𝐴, ∅⦆ = ⦅𝐴. Note that in the context of Morse tuples, it is natural to define the 0-tuple as the empty set. Therefore, the present theorem together with bj-1upln0 32021 and bj-2upln0 32035 tell us that an m-tuple may equal an n-tuple only when m = n, at least for m, n <= 2, but this result would extend as soon as we define n-tuples for higher values of n. (Contributed by BJ, 21-Apr-2019.)
𝐴, 𝐵⦆ ≠ ⦅𝐶

20.14.5.14  Set theory: miscellaneous

Miscellaneous theorems of set theory.

Theorembj-vjust2 32037 Justification theorem for bj-df-v 32038. See also vjust 3078 and bj-vjust 31815. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
{𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}

Theorembj-df-v 32038 Alternate definition of the universal class. Actually, the current definition df-v 3079 should be proved from this one, and vex 3080 should be proved from this proposed definition together with bj-vexwv 31882, which would remove from vex 3080 dependency on ax-13 2137 (see also comment of bj-vexw 31880). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
V = {𝑥 ∣ ⊤}

Theorembj-df-nul 32039 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = {𝑥 ∣ ⊥}

Theorembj-nul 32040* Two formulations of the axiom of the empty set ax-nul 4616. Proposal: place it right before ax-nul 4616. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
(∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliota 32041* Definition of the empty set using the definite description binder. See also bj-nuliotaALT 32042. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-nuliotaALT 32042* Alternate proof of bj-nuliota 32041. Note that this alternate proof uses the fact that 𝑥𝜑 evaluates to when there is no 𝑥 satisfying 𝜑 (iotanul 5668). This is an implementation detail of the encoding currently used in set.mm and should be avoided. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = (℩𝑥𝑦 ¬ 𝑦𝑥)

Theorembj-vtoclgfALT 32043 Alternate proof of vtoclgf 3141. Proof from vtoclgft 3131. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theorembj-pwcfsdom 32044 Remove hypothesis from pwcfsdom 9160. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 9160.) (Contributed by BJ, 14-Sep-2019.)
(ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))

Theorembj-grur1 32045 Remove hypothesis from grur1 9397. Illustration of how to remove a "definitional hypothesis". This makes its uses longer, but the theorem feels more self-contained. Looks preferable when the defined term appears only once in the conclusion. (Contributed by BJ, 14-Sep-2019.)
((𝑈 ∈ Univ ∧ 𝑈 (𝑅1 “ On)) → 𝑈 = (𝑅1‘(𝑈 ∩ On)))

20.14.5.15  Elementwise intersection (families of sets induced on a subset)

Many kinds of structures are given by families of subsets of a given set: Moore collections (df-mre 15961), topologies (df-top 20424), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 29294), sigma rings, monotone classes, matroids/independent sets, bornologies, filters.

There is a natural notion of structure induced on a subset. It is often given by an elementwise intersection, namely, the family of intersections of sets in the original family with the given subset. In this subsection, we define this notion and prove its main properties. Classical conditions on families of subsets include being nonempty, containing the whole set, containing the empty set, being stable under unions, intersections, subsets, supersets, (relative) complements. Therefore, we prove related properties for the elementwise intersection.

We will call (𝑋t 𝐴) the elementwise intersection on the family 𝑋 by the class 𝐴.

REMARK: many theorems are already in set.mm ; MM>search *rest* /J

Theorembj-rest00 32046 An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 15797. (Contributed by BJ, 27-Apr-2021.)
(∅ ↾t 𝐴) = ∅

Theorembj-restsn 32047 An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 32050 and bj-restsnid 32052. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})

Theorembj-restsnss 32048 Special case of bj-restsn 32047. (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})

Theorembj-restsnss2 32049 Special case of bj-restsn 32047. (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝑌𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})

Theorembj-restsn0 32050 An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 32047 and bj-restsnss2 32049. TODO: this is restsn 20687. (Contributed by BJ, 27-Apr-2021.)
(𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Theorembj-restsn10 32051 Special case of bj-restsn 32047, bj-restsnss 32048, and bj-rest10 32053. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ({𝑋} ↾t ∅) = {∅})

Theorembj-restsnid 32052 The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 32047 and bj-restsnss 32048. (Contributed by BJ, 27-Apr-2021.)
({𝐴} ↾t 𝐴) = {𝐴}

Theorembj-rest10 32053 An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 20686 and could replace it. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝑋 ≠ ∅ → (𝑋t ∅) = {∅}))

Theorembj-rest10b 32054 Alternate version of bj-rest10 32053. (Contributed by BJ, 27-Apr-2021.)
(𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋t ∅) = {∅})

Theorembj-restn0 32055 An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))

Theorembj-restn0b 32056 Alternate version of bj-restn0 32055. (Contributed by BJ, 27-Apr-2021.)
((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴𝑊) → (𝑋t 𝐴) ≠ ∅)

Theorembj-restpw 32057 The elementwise intersection on a powerset is the powerset of the intersection. This allows to prove for instance that the topology induced on a subset by the discrete topology is the discrete topology on that subset. See also restdis 20695 (which uses distop 20513 and restopn2 20694). (Contributed by BJ, 27-Apr-2021.)
((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))

Theorembj-rest0 32058 An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋t 𝐴)))

Theorembj-restb 32059 An elementwise intersection by a set on a family containing a superset of that set contains that set. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → ((𝐴𝐵𝐵𝑋) → 𝐴 ∈ (𝑋t 𝐴)))

Theorembj-restv 32060 An elementwise intersection by a subset on a family containing the whole set contains the whole subset. (Contributed by BJ, 27-Apr-2021.)
((𝐴 𝑋 𝑋𝑋) → 𝐴 ∈ (𝑋t 𝐴))

Theorembj-resta 32061 An elementwise intersection by a set on a family containing that set contains that set. (Contributed by BJ, 27-Apr-2021.)
(𝑋𝑉 → (𝐴𝑋𝐴 ∈ (𝑋t 𝐴)))

Theorembj-restuni 32062 The union of an elementwise intersection by a set is equal to the intersection with that set of the union of the family. See also restuni 20679 and restuni2 20684. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))

Theorembj-restuni2 32063 The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 20679 and restuni2 20684. (Contributed by BJ, 27-Apr-2021.)
((𝑋𝑉𝐴 𝑋) → (𝑋t 𝐴) = 𝐴)

Theorembj-restreg 32064 A reformulation of the axiom of regularity using elementwise intersection. (RK: might have to be placed later since theorems in this section are to be moved early (in the section related to the algebra of sets).) (Contributed by BJ, 27-Apr-2021.)
((𝐴𝑉𝐴 ≠ ∅) → ∅ ∈ (𝐴t 𝐴))

20.14.5.16  Topology (complements)

Theorembj-toptopon2 32065 A topology is the same thing as a topology on the union of its open sets. space. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))

Theorembj-topontopon 32066 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))

Theorembj-funtopon 32067 TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn

Theorembj-elpw3 32068 A variant of elpwg 4019. (Contributed by BJ, 29-Apr-2021.)
((𝐴 ∈ V ∧ 𝐴𝐵) ↔ 𝐴 ∈ 𝒫 𝐵)

Theorembj-sspwpw 32069 The union of a set is included in a given class if and only if that set is an element of the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
((𝐴 ∈ V ∧ 𝐴𝐵) ↔ 𝐴 ∈ 𝒫 𝒫 𝐵)

Theorembj-sspwpwab 32070* The class of families whose union is included in a given class is equal to the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
{𝑥 𝑥𝐴} = 𝒫 𝒫 𝐴

Theorembj-sspwpweq 32071* The class of families whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
{𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴

Theorembj-toponss 32072 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.)
(TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴

Theorembj-dmtopon 32073 The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V

Theorembj-fntopon 32074 TopOn is a function with domain V. Analogue of fnmre 15966. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V

Theorembj-toprntopon 32075 A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)
Top = ran TopOn

Theorembj-xnex 32076* Lemma for snnex 6738 and bj-pwnex 32077. (Contributed by BJ, 2-May-2021.)
(∀𝑦(𝐴𝑉𝑦𝐴) → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V)

Theorembj-pwnex 32077* The class of all power sets is a proper class. See also snnex 6738. (Contributed by BJ, 2-May-2021.)
{𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Theorembj-topnex 32078 The class of all topologies is a proper class. (Contributed by BJ, 2-May-2021.)
Top ∉ V

Syntaxcmoo 32079 Syntax for the class of Moore collections.
class Moore

Definitiondf-bj-mre 32080* Define the class of Moore collections. This is to df-mre 15961 what df-top 20424 is to df-topon 20426.

Note: df-mre 15961 singles out the empty intersection. This is not necessary. It could be written instead Moore = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝒫 𝑥 ∣ ∀𝑧 ∈ 𝒫 𝑦(𝑥 𝑧) ∈ 𝑦})

Remark: as usual, if one wanted a more general definition, one could define a new syntax and (Moore𝐴 ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴). (Contributed by BJ, 27-Apr-2021.)

Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 𝑦) ∈ 𝑥}

20.14.5.17  Maps-to notation for functions with three arguments

Theorembj-0nelmpt 32081 The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)
¬ ∅ ∈ (𝑥𝐴𝐵)

Theorembj-mptval 32082 Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020.)
𝑥𝐴       (∀𝑥𝐴 𝐵𝑉 → (𝑋𝐴 → (((𝑥𝐴𝐵)‘𝑋) = 𝑌𝑋(𝑥𝐴𝐵)𝑌)))

Theorembj-dfmpt2a 32083* An equivalent definition of df-mpt2 6431. (Contributed by BJ, 30-Dec-2020.)
(𝑥𝐴, 𝑦𝐵𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}

Theorembj-mpt2mptALT 32084* Alternate proof of mpt2mpt 6527. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Syntaxcmpt3 32085 Extend the definition of a class to include maps-to notation for functions with three arguments.
class (𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷)

Definitiondf-bj-mpt3 32086* Define maps-to notation for functions with three arguments. See df-mpt 4543 and df-mpt2 6431 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpt2a 32083. (Contributed by BJ, 11-Apr-2020.)
(𝑥𝐴, 𝑦𝐵, 𝑧𝐶𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵𝑧𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}

20.14.5.18  Currying

Currying and uncurrying. See also df-cur and df-unc 7156. Contrary to these, the definitions in this section are parameterized.

Syntaxcfset 32087 Notation for the set of functions between two sets.
class Set

Definitiondf-bj-fset 32088* Define the set of functions between two sets. Same as df-map 7622 with arguments swapped. TODO: prove the same staple lemmas as for 𝑚. (Contributed by BJ, 11-Apr-2020.)
Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑥𝑦})

Syntaxccur- 32089 Extend class notation to include the parameterized currying function.
class curry_

Syntaxcunc- 32090 Extend class notation to include the parameterized uncurrying function.
class uncurry_

Definitiondf-bj-cur 32091* Define currying. See also df-cur 7155. (Contributed by BJ, 11-Apr-2020.)
curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set𝑧) ↦ (𝑎𝑥 ↦ (𝑏𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))

Definitiondf-bj-unc 32092* Define uncurrying. See also df-unc 7156. (Contributed by BJ, 11-Apr-2020.)
uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set𝑧)) ↦ (𝑎𝑥, 𝑏𝑦 ↦ ((𝑓𝑎)‘𝑏))))

20.14.6  Extended real and complex numbers, real and complex projectives lines

In this section, we indroduce several supersets of the set of real numbers and the set of complex numbers.

Once they are given their usual topologies, which are locally compact, both topological spaces have a one-point compactification. They are denoted by ℝ̂ and ℂ̂ respectively, defined in df-bj-cchat 32128 and df-bj-rrhat 32130, and the point at infinity is denoted by , defined in df-bj-infty 32126.

Both and also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 32124 (already defined as *, see df-xr 9833) and ℂ̅, defined in df-bj-ccbar 32111.

Since ℂ̅ does not seem to be standard, we describe it in some detail. It is obtained by adding to a "point at infinity at the end of each ray with origin at 0". Although ℂ̅ is not an important object in itself, the motivation for introducing it is to provide a common superset to both ℝ̅ and and to define algebraic operations (addition, opposite, multiplication, inverse) as widely as reasonably possible.

Mathematically, ℂ̅ is the quotient of ((ℂ × ℝ≥0) ∖ {⟨0, 0⟩}) by the diagonal multiplicative action of >0 (think of the closed "northern hemisphere" in ^3 identified with (ℂ × ℝ), that each open ray from 0 included in the closed northern half-space intersects exactly once).

Since in set.mm, we want to have a genuine inclusion ℂ ⊆ ℂ̅, we instead define ℂ̅ as the (disjoint) union of with a circle at infinity denoted by . To have a genuine inclusion ℝ̅ ⊆ ℂ̅, we define +∞ and -∞ as certain points in .

Thanks to this framework, one has the genuine inclusions ℝ ⊆ ℝ̅ and ℝ ⊆ ℝ̂ and similarly ℂ ⊆ ℂ̅ and ℂ ⊆ ℂ̂. Furthermore, one has ℝ ⊆ ℂ as well as ℝ̅ ⊆ ℂ̅ and ℝ̂ ⊆ ℂ̂.

Furthermore, we define the main algebraic operations on (ℂ̅ ∪ ℂ̂), which is not very mathematical, but "overloads" the operations, so that one can use the same notation in all cases.

20.14.6.1  Diagonal in a Cartesian square

Complements on the idendity relation and definition of the diagonal in the Cartesian square of a set.

Theorembj-elid 32093 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))

Theorembj-elid2 32094 Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
(𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))

Theorembj-elid3 32095 Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)
(⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Syntaxcdiag2 32096 Syntax for the diagonal of the Cartesian square of a set.
class Diag

Definitiondf-bj-diag 32097 Define the diagonal of the Cartesian square of a set. (Contributed by BJ, 22-Jun-2019.)
Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))

Theorembj-diagval 32098 Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))

Theorembj-eldiag 32099 Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st𝐵) = (2nd𝐵))))

Theorembj-eldiag2 32100 Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
(𝐴𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵𝐴𝐵 = 𝐶)))

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