P. 323 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-pr2ex 32201	Sethood of the second projection. (Contributed by BJ, 6-Oct-2018.)
⊢ (𝐴 ∈ 𝑉 → pr2 𝐴 ∈ V)
Theorem	bj-2uplth 32202	The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 4871). (Contributed by BJ, 6-Oct-2018.)
⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	bj-2uplex 32203	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))
Theorem	bj-2upln0 32204	A couple is nonempty. (Contributed by BJ, 21-Apr-2019.)
⊢ ⦅𝐴, 𝐵⦆ ≠ ∅
Theorem	bj-2upln1upl 32205	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 32190 and bj-2upln0 32204 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⦆
21.14.5.14  Set theory: miscellaneous Miscellaneous theorems of set theory.
Theorem	bj-vjust2 32206	Justification theorem for bj-df-v 32207. See also vjust 3174 and bj-vjust 31974. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤}
Theorem	bj-df-v 32207	Alternate definition of the universal class. Actually, the current definition df-v 3175 should be proved from this one, and vex 3176 should be proved from this proposed definition together with bj-vexwv 32051, which would remove from vex 3176 dependency on ax-13 2234 (see also comment of bj-vexw 32049). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ V = {𝑥 ∣ ⊤}
Theorem	bj-df-nul 32208	Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ ∅ = {𝑥 ∣ ⊥}
Theorem	bj-nul 32209*	Two formulations of the axiom of the empty set ax-nul 4717. Proposal: place it right before ax-nul 4717. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-nuliota 32210*	Definition of the empty set using the definite description binder. See also bj-nuliotaALT 32211. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-nuliotaALT 32211*	Alternate proof of bj-nuliota 32210. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 5783). 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.)
⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	bj-vtoclgfALT 32212	Alternate proof of vtoclgf 3237. Proof from vtoclgft 3227. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)
Theorem	bj-pwcfsdom 32213	Remove hypothesis from pwcfsdom 9284. Illustration of how to remove a "proof-facilitating hypothesis". (Can use it to shorten theorems using pwcfsdom 9284.) (Contributed by BJ, 14-Sep-2019.)
⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
Theorem	bj-grur1 32214	Remove hypothesis from grur1 9521. 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)))
21.14.5.15  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 16069), topologies (df-top 20521), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga 29498), 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 (𝑋 ↾t 𝐴) the elementwise intersection on the family 𝑋 by the class 𝐴. REMARK: many theorems are already in set.mm ; MM>search *rest* /J
Theorem	bj-rest00 32215	An elementwise intersection on the empty family is the empty set. TODO: this is 0rest 15913. (Contributed by BJ, 27-Apr-2021.)
⊢ (∅ ↾t 𝐴) = ∅
Theorem	bj-restsn 32216	An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 32219 and bj-restsnid 32221. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)})
Theorem	bj-restsnss 32217	Special case of bj-restsn 32216. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌) → ({𝑌} ↾t 𝐴) = {𝐴})
Theorem	bj-restsnss2 32218	Special case of bj-restsn 32216. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → ({𝑌} ↾t 𝐴) = {𝑌})
Theorem	bj-restsn0 32219	An elementwise intersection on the singleton on the empty set is the singleton on the empty set. Special case of bj-restsn 32216 and bj-restsnss2 32218. TODO: this is restsn 20784. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝐴 ∈ 𝑉 → ({∅} ↾t 𝐴) = {∅})
Theorem	bj-restsn10 32220	Special case of bj-restsn 32216, bj-restsnss 32217, and bj-rest10 32222. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → ({𝑋} ↾t ∅) = {∅})
Theorem	bj-restsnid 32221	The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 32216 and bj-restsnss 32217. (Contributed by BJ, 27-Apr-2021.)
⊢ ({𝐴} ↾t 𝐴) = {𝐴}
Theorem	bj-rest10 32222	An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 20783 and could replace it. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝑋 ≠ ∅ → (𝑋 ↾t ∅) = {∅}))
Theorem	bj-rest10b 32223	Alternate version of bj-rest10 32222. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ (𝑉 ∖ {∅}) → (𝑋 ↾t ∅) = {∅})
Theorem	bj-restn0 32224	An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ≠ ∅ → (𝑋 ↾t 𝐴) ≠ ∅))
Theorem	bj-restn0b 32225	Alternate version of bj-restn0 32224. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ (𝑉 ∖ {∅}) ∧ 𝐴 ∈ 𝑊) → (𝑋 ↾t 𝐴) ≠ ∅)
Theorem	bj-restpw 32226	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 20792 (which uses distop 20610 and restopn2 20791). (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))
Theorem	bj-rest0 32227	An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋 ↾t 𝐴)))
Theorem	bj-restb 32228	An elementwise intersection by a set on a family containing a superset of that set contains that set. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴)))
Theorem	bj-restv 32229	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 ∧ ∪ 𝑋 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴))
Theorem	bj-resta 32230	An elementwise intersection by a set on a family containing that set contains that set. (Contributed by BJ, 27-Apr-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝐴 ∈ 𝑋 → 𝐴 ∈ (𝑋 ↾t 𝐴)))
Theorem	bj-restuni 32231	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 20776 and restuni2 20781. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))
Theorem	bj-restuni2 32232	The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 20776 and restuni2 20781. (Contributed by BJ, 27-Apr-2021.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ ∪ 𝑋) → ∪ (𝑋 ↾t 𝐴) = 𝐴)
Theorem	bj-restreg 32233	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∅ ∈ (𝐴 ↾t 𝐴))
21.14.5.16  Topology (complements)
Theorem	bj-toptopon2 32234	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))
Theorem	bj-topontopon 32235	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))
Theorem	bj-funtopon 32236	TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
⊢ Fun TopOn
Theorem	bj-elpw3 32237	A variant of elpwg 4116. (Contributed by BJ, 29-Apr-2021.)
⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ∈ 𝒫 𝐵)
Theorem	bj-sspwpw 32238	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 ∧ ∪ 𝐴 ⊆ 𝐵) ↔ 𝐴 ∈ 𝒫 𝒫 𝐵)
Theorem	bj-sspwpwab 32239*	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 ∣ ∪ 𝑥 ⊆ 𝐴} = 𝒫 𝒫 𝐴
Theorem	bj-sspwpweq 32240*	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
Theorem	bj-toponss 32241	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
Theorem	bj-dmtopon 32242	The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.)
⊢ dom TopOn = V
Theorem	bj-fntopon 32243	TopOn is a function with domain V. Analogue of fnmre 16074. (Contributed by BJ, 29-Apr-2021.)
⊢ TopOn Fn V
Theorem	bj-toprntopon 32244	A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)
⊢ Top = ∪ ran TopOn
Theorem	bj-xnex 32245*	Lemma for snnex 6862 and bj-pwnex 32246. (Contributed by BJ, 2-May-2021.)
⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V)
Theorem	bj-pwnex 32246*	The class of all power sets is a proper class. See also snnex 6862. (Contributed by BJ, 2-May-2021.)
⊢ {𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V
Theorem	bj-topnex 32247	The class of all topologies is a proper class. (Contributed by BJ, 2-May-2021.)
⊢ Top ∉ V
Syntax	cmoo 32248	Syntax for the class of Moore collections.
class Moore
Definition	df-bj-mre 32249*	Define the class of Moore collections. This is to df-mre 16069 what df-top 20521 is to df-topon 20523. Note: df-mre 16069 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}
21.14.5.17  Maps-to notation for functions with three arguments
Theorem	bj-0nelmpt 32250	The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)
⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	bj-mptval 32251	Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑋 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑋) = 𝑌 ↔ 𝑋(𝑥 ∈ 𝐴 ↦ 𝐵)𝑌)))
Theorem	bj-dfmpt2a 32252*	An equivalent definition of df-mpt2 6554. (Contributed by BJ, 30-Dec-2020.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
Theorem	bj-mpt2mptALT 32253*	Alternate proof of mpt2mpt 6650. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)    ⇒   ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
Syntax	cmpt3 32254	Extend the definition of a class to include maps-to notation for functions with three arguments.
class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
Definition	df-bj-mpt3 32255*	Define maps-to notation for functions with three arguments. See df-mpt 4645 and df-mpt2 6554 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpt2a 32252. (Contributed by BJ, 11-Apr-2020.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑠 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝑡 = 𝐷)}
21.14.5.18  Currying Currying and uncurrying. See also df-cur and df-unc 7281. Contrary to these, the definitions in this section are parameterized.
Syntax	cfset 32256	Notation for the set of functions between two sets.
class Set⟶
Definition	df-bj-fset 32257*	Define the set of functions between two sets. Same as df-map 7746 with arguments swapped. TODO: prove the same staple lemmas as for ↑𝑚. (Contributed by BJ, 11-Apr-2020.)
⊢ Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦})
Syntax	ccur- 32258	Extend class notation to include the parameterized currying function.
class curry_
Syntax	cunc- 32259	Extend class notation to include the parameterized uncurrying function.
class uncurry_
Definition	df-bj-cur 32260*	Define currying. See also df-cur 7280. (Contributed by BJ, 11-Apr-2020.)
⊢ curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘⟨𝑎, 𝑏⟩)))))
Definition	df-bj-unc 32261*	Define uncurrying. See also df-unc 7281. (Contributed by BJ, 11-Apr-2020.)
⊢ uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏))))
21.14.6  Extended real and complex numbers, real and complex projectives lines In this section, we indroduce several supersets of the set ℝ of real numbers and the set ℂ 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 32297 and df-bj-rrhat 32299, and the point at infinity is denoted by ∞, defined in df-bj-infty 32295. Both ℝ and ℂ also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 32293 (already defined as ℝ*, see df-xr 9957) and ℂ̅, defined in df-bj-ccbar 32280. Since ℂ̅ does not seem to be standard, we describe it in some detail. It is obtained by adding to ℂ 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 ℂ 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 ℝ>0 (think of the closed "northern hemisphere" in ℝ^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 ℂ with a circle at infinity denoted by ℂ∞. To have a genuine inclusion ℝ̅ ⊆ ℂ̅, we define +∞ and -∞ as certain points in ℂ∞. 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.
21.14.6.1  Diagonal in a Cartesian square Complements on the idendity relation and definition of the diagonal in the Cartesian square of a set.
Theorem	bj-elid 32262	Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))
Theorem	bj-elid2 32263	Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
Theorem	bj-elid3 32264	Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)
⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Syntax	cdiag2 32265	Syntax for the diagonal of the Cartesian square of a set.
class Diag
Definition	df-bj-diag 32266	Define the diagonal of the Cartesian square of a set. (Contributed by BJ, 22-Jun-2019.)
⊢ Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥)))
Theorem	bj-diagval 32267	Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (Diag‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Theorem	bj-eldiag 32268	Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Diag‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))
Theorem	bj-eldiag2 32269	Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Diag‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)))
21.14.6.2  Extended numbers and projective lines as sets TODO(?): replace df-bj-inftyexpi 32271 with a function inftyexpi2pi defined on (0[,)1) since we plan to put this section as early as possible, before the definition of π. It would be best to use df-0r 9761 and df-1r 9762 but intervals are defined for real numers, and not these temporary reals. It looks like to define the sets, the addition and the opposite, one only needs some basic results about addition, opposite and ordering, which could use df-plr 9758, df-ltr 9760, df-0r 9761, df-1r 9762, df-ltr 9760. The idea is then to define the order relation directly on ℝ̅, skipping ℝ.
Syntax	cinftyexpi 32270	Syntax for the function inftyexpi parameterizing ℂ∞.
class inftyexpi
Definition	df-bj-inftyexpi 32271	Definition of the auxiliary function inftyexpi parameterizing the circle at infinity ℂ∞ in ℂ̅. We use coupling with ℂ to simplify the proof of bj-ccinftydisj 32277. It could seem more natural to define inftyexpi on all of ℝ using prcpal but we want to use only basic functions in the definition of ℂ̅. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
⊢ inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
Theorem	bj-inftyexpiinv 32272	Utility theorem for the inverse of inftyexpi. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
Theorem	bj-inftyexpiinj 32273	Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 32272 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.)
⊢ ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))
Theorem	bj-inftyexpidisj 32274	An element of the circle at infinity is not a complex number. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
⊢ ¬ (inftyexpi ‘𝐴) ∈ ℂ
Syntax	cccinfty 32275	Syntax for the circle at infinity ℂ∞.
class ℂ∞
Definition	df-bj-ccinfty 32276	Definition of the circle at infinity ℂ∞. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
⊢ ℂ∞ = ran inftyexpi
Theorem	bj-ccinftydisj 32277	The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
⊢ (ℂ ∩ ℂ∞) = ∅
Theorem	bj-elccinfty 32278	A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) ∈ ℂ∞)
Syntax	cccbar 32279	Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅
Definition	df-bj-ccbar 32280	Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
⊢ ℂ̅ = (ℂ ∪ ℂ∞)
Theorem	bj-ccssccbar 32281	Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ ⊆ ℂ̅
Theorem	bj-ccinftyssccbar 32282	Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ∞ ⊆ ℂ̅
Syntax	cpinfty 32283	Syntax for +∞.
class +∞
Definition	df-bj-pinfty 32284	Definition of +∞. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ = (inftyexpi ‘0)
Theorem	bj-pinftyccb 32285	The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ∈ ℂ̅
Theorem	bj-pinftynrr 32286	The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ +∞ ∈ ℂ
Syntax	cminfty 32287	Syntax for -∞.
class -∞
Definition	df-bj-minfty 32288	Definition of -∞. (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ = (inftyexpi ‘π)
Theorem	bj-minftyccb 32289	The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ ∈ ℂ̅
Theorem	bj-minftynrr 32290	The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ -∞ ∈ ℂ
Theorem	bj-pinftynminfty 32291	The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ≠ -∞
Syntax	crrbar 32292	Syntax for the set of extended real numbers ℝ̅.
class ℝ̅
Definition	df-bj-rrbar 32293	Definition of the set of extended real numbers ℝ̅. See df-xr 9957. (Contributed by BJ, 29-Jun-2019.)
⊢ ℝ̅ = (ℝ ∪ {-∞, +∞})
Syntax	cinfty 32294	Syntax for ∞.
class ∞
Definition	df-bj-infty 32295	Definition of ∞, the point at infinity of the real or complex projective line. (Contributed by BJ, 27-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
⊢ ∞ = 𝒫 ∪ ℂ
Syntax	ccchat 32296	Syntax for ℂ̂.
class ℂ̂
Definition	df-bj-cchat 32297	Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ̂ = (ℂ ∪ {∞})
Syntax	crrhat 32298	Syntax for ℝ̂.
class ℝ̂
Definition	df-bj-rrhat 32299	Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
⊢ ℝ̂ = (ℝ ∪ {∞})
Theorem	bj-rrhatsscchat 32300	The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.)
⊢ ℝ̂ ⊆ ℂ̂