P. 372 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37101-37200   *Has distinct variable group(s)

Type	Label	Description
Statement
21.19.5.22  Moore collections (complements)
Theorem	bj-raldifsn 37101*	All elements in a set satisfy a given property if and only if all but one satisfy that property and that one also does. Typically, this can be used for characterizations that are proved using different methods for a given element and for all others, for instance zero and nonzero numbers, or the empty set and nonempty sets. (Contributed by BJ, 7-Dec-2021.)
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ∧ 𝜓)))
Theorem	bj-0int 37102*	If 𝐴 is a collection of subsets of 𝑋, like a Moore collection or a topology, two equivalent ways to say that arbitrary intersections of elements of 𝐴 relative to 𝑋 belong to some class 𝐵: the LHS singles out the empty intersection (the empty intersection relative to 𝑋 is 𝑋 and the intersection of a nonempty family of subsets of 𝑋 is included in 𝑋, so there is no need to intersect it with 𝑋). In typical applications, 𝐵 is 𝐴 itself. (Contributed by BJ, 7-Dec-2021.)
⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵))
Theorem	bj-mooreset 37103*	A Moore collection is a set. Therefore, the class Moore of all Moore sets defined in df-bj-moore 37105 is actually the class of all Moore collections. This is also illustrated by the lack of sethood condition in bj-ismoore 37106. Note that the closed sets of a topology form a Moore collection, so a topology is a set, and this remark also applies to many other families of sets (namely, as soon as the whole set is required to be a set of the family, then the associated kind of family has no proper classes: that this condition suffices to impose sethood can be seen in this proof, which relies crucially on uniexr 7783). Note: if, in the above predicate, we substitute 𝒫 𝑋 for 𝐴, then the last ∈ 𝒫 𝑋 could be weakened to ⊆ 𝑋, and then the predicate would be obviously satisfied since ⊢ ∪ 𝒫 𝑋 = 𝑋 (unipw 5455), making 𝒫 𝑋 a Moore collection in this weaker sense, for any class 𝑋, even proper, but the addition of this single case does not add anything interesting. Instead, we have the biconditional bj-discrmoore 37112. (Contributed by BJ, 8-Dec-2021.)
⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)
Syntax	cmoore 37104	Syntax for the class of Moore collections.
class Moore
Definition	df-bj-moore 37105*	Define the class of Moore collections. This is indeed the class of all Moore collections since these all are sets, as proved in bj-mooreset 37103, and as illustrated by the lack of sethood condition in bj-ismoore 37106. This is to df-mre 17629 (defining Moore) what df-top 22900 (defining Top) is to df-topon 22917 (defining TopOn). For the sake of consistency, the function defined at df-mre 17629 should be denoted by "MooreOn". Note: df-mre 17629 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}) and the equivalence of both definitions is proved by bj-0int 37102. There is no added generality in defining a "Moore predicate" for arbitrary classes, since a Moore class satisfying such a predicate is automatically a set (see bj-mooreset 37103). TODO: move to the main section. For many families of sets, one can define both the function associating to each set the set of families of that kind on it (like df-mre 17629 and df-topon 22917) or the class of all families of that kind, independent of a base set (like df-bj-moore 37105 or df-top 22900). In general, the former will be more useful and the extra generality of the latter is not necessary. Moore collections, however, are particular in that they are more ubiquitous and are used in a wide variety of applications (for many families of sets, the family of families of a given kind is often a Moore collection, for instance). Therefore, in the case of Moore families, having both definitions is useful. (Contributed by BJ, 27-Apr-2021.)
⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦) ∈ 𝑥}
Theorem	bj-ismoore 37106*	Characterization of Moore collections. Note that there is no sethood hypothesis on 𝐴: it is implied by either side (this is obvious for the LHS, and is the content of bj-mooreset 37103 for the RHS). (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)
Theorem	bj-ismoored0 37107	Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴)
Theorem	bj-ismoored 37108	Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ Moore)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴)
Theorem	bj-ismoored2 37109	Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝜑 → 𝐴 ∈ Moore)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    ⇒   ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴)
Theorem	bj-ismooredr 37110*	Sufficient condition to be a Moore collection. Note that there is no sethood hypothesis on 𝐴: it is a consequence of the only hypothesis. (Contributed by BJ, 9-Dec-2021.)
⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ∈ Moore)
Theorem	bj-ismooredr2 37111*	Sufficient condition to be a Moore collection (variant of bj-ismooredr 37110 singling out the empty intersection). Note that there is no sethood hypothesis on 𝐴: it is a consequence of the first hypothesis. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) → ∩ 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ∈ Moore)
Theorem	bj-discrmoore 37112	The powerclass 𝒫 𝐴 is a Moore collection if and only if 𝐴 is a set. It is then called the discrete Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore)
Theorem	bj-0nmoore 37113	The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.)
⊢ ¬ ∅ ∈ Moore
Theorem	bj-snmoore 37114	A singleton is a Moore collection. See bj-snmooreb 37115 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)
Theorem	bj-snmooreb 37115	A singleton is a Moore collection, biconditional version. (Contributed by BJ, 9-Dec-2021.) (Proof shortened by BJ, 10-Apr-2024.)
⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore)
Theorem	bj-prmoore 37116	A pair formed of two nested sets is a Moore collection. (Note that in the statement, if 𝐵 is a proper class, we are in the case of bj-snmoore 37114). A direct consequence is ⊢ {∅, 𝐴} ∈ Moore. More generally, any nonempty well-ordered chain of sets that is a set is a Moore collection. We also have the biconditional ⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → ({𝐴, 𝐵} ∈ Moore ↔ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))). (Contributed by BJ, 11-Apr-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → {𝐴, 𝐵} ∈ Moore)
21.19.5.23  Maps-to notation for functions with three arguments
Theorem	bj-0nelmpt 37117	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 37118	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-dfmpoa 37119*	An equivalent definition of df-mpo 7436. (Contributed by BJ, 30-Dec-2020.)
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨𝑠, 𝑡⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑠 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶)}
Theorem	bj-mpomptALT 37120*	Alternate proof of mpompt 7547. (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 37121	Syntax for maps-to notation for functions with three arguments.
class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
Definition	df-bj-mpt3 37122*	Define maps-to notation for functions with three arguments. See df-mpt 5226 and df-mpo 7436 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpoa 37119. (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)}
21.19.5.24  Currying Currying and uncurrying. See also df-cur 8292 and df-unc 8293. Contrary to these, the definitions in this section are parameterized.
Syntax	csethom 37123	Syntax for the set of set morphisms.
class Set⟶
Definition	df-bj-sethom 37124*	Define the set of functions (morphisms of sets) between two sets. Same as df-map 8868 with arguments swapped. TODO: prove the same staple lemmas as for ↑m. Remark: one may define Set⟶ = (𝑥 ∈ dom Struct , 𝑦 ∈ dom Struct ↦ {𝑓 ∣ 𝑓:(Base‘𝑥)⟶(Base‘𝑦)}) so that for morphisms between other structures, one could write ... = {𝑓 ∈ (𝑥 Set⟶ 𝑦) ∣ ...}. (Contributed by BJ, 11-Apr-2020.)
⊢ Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦})
Syntax	ctophom 37125	Syntax for the set of topological morphisms.
class Top⟶
Definition	df-bj-tophom 37126*	Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 23235 (which is in terms of topologies instead of topological spaces). (Contributed by BJ, 10-Feb-2022.)
⊢ Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(◡𝑓 “ 𝑢) ∈ (TopOpen‘𝑥)})
Syntax	cmgmhom 37127	Syntax for the set of magma morphisms.
class Mgm⟶
Definition	df-bj-mgmhom 37128*	Define the set of magma morphisms between two magmas. If domain and codomain are semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. (Contributed by BJ, 10-Feb-2022.)
⊢ Mgm⟶ = (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))})
Syntax	ctopmgmhom 37129	Syntax for the set of topological magma morphisms.
class TopMgm⟶
Definition	df-bj-topmgmhom 37130*	Define the set of topological magma morphisms (continuous magma morphisms) between two topological magmas. If domain and codomain are topological semigroups, monoids, or groups, then one obtains the set of morphisms of these structures. This definition is currently stated with topological monoid domain and codomain, since topological magmas are currently not defined in set.mm. (Contributed by BJ, 10-Feb-2022.)
⊢ TopMgm⟶ = (𝑥 ∈ TopMnd, 𝑦 ∈ TopMnd ↦ ((𝑥 Top⟶ 𝑦) ∩ (𝑥 Mgm⟶ 𝑦)))
Syntax	ccur- 37131	Syntax for the parameterized currying function.
class curry_
Definition	df-bj-cur 37132*	Define currying. See also df-cur 8292. (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⟩)))))
Syntax	cunc- 37133	Notation for the parameterized uncurrying function.
class uncurry_
Definition	df-bj-unc 37134*	Define uncurrying. See also df-unc 8293. (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))))
21.19.5.25  Setting components of extensible structures Groundwork for changing the definition, syntax and token for component-setting in extensible structures. See https://github.com/metamath/set.mm/issues/2401
Syntax	cstrset 37135	Syntax for component-setting in extensible structures.
class [𝐵 / 𝐴]struct𝑆
Definition	df-strset 37136	Component-setting in extensible structures. Define the extensible structure [𝐵 / 𝐴]struct𝑆, which is like the extensible structure 𝑆 except that the value 𝐵 has been put in the slot 𝐴 (replacing the current value if there was already one). In such expressions, 𝐴 is generally substituted for slot mnemonics like Base or +g or dist. The V in this definition was chosen to be closer to df-sets 17201, but since extensible structures are functions on ℕ, it will be more natural to replace it with ℕ when df-strset 37136 becomes the main definition. (Contributed by BJ, 13-Feb-2022.)
⊢ [𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
Theorem	setsstrset 37137	Relation between df-sets 17201 and df-strset 37136. Temporary theorem kept during the transition from the former to the latter. (Contributed by BJ, 13-Feb-2022.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → [𝐵 / 𝐴]struct𝑆 = (𝑆 sSet ⟨(𝐴‘ndx), 𝐵⟩))
21.19.6  Extended real and complex numbers, real and complex projective 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 37234 and df-bj-rrhat 37236, and the point at infinity is denoted by ∞, defined in df-bj-infty 37232. Both ℝ and ℂ also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 37230 (already defined as ℝ*, see df-xr 11299) and ℂ̅, defined in df-bj-ccbar 37217. 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.
21.19.6.1  Complements on class abstractions of ordered pairs and binary relations
Theorem	bj-nfald 37138	Variant of nfald 2328. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	bj-nfexd 37139	Variant of nfexd 2329. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
Theorem	copsex2d 37140*	Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
Theorem	copsex2b 37141*	Biconditional form of copsex2d 37140. TODO: prove a relative version, that is, with ∃𝑥 ∈ 𝑉∃𝑦 ∈ 𝑊...(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊). (Contributed by BJ, 27-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Theorem	opelopabd 37142*	Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
Theorem	opelopabb 37143*	Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Theorem	opelopabbv 37144*	Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Theorem	bj-opelrelex 37145	The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5737, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)
⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-opelresdm 37146	If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 6003. (Contributed by BJ, 25-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
Theorem	bj-brresdm 37147	If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 6004 and brrelex1 5738. Remark: there are many pairs like bj-opelresdm 37146 / bj-brresdm 37147, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 37146 / brrelex12 5737 or the opelopabg 5543 / brabg 5544 family). They are straightforwardly equivalent by df-br 5144. The latter is indeed a very direct definition, introducing a "shorthand", and barely necessary, were it not for the frequency of the expression 𝐴𝑅𝐵. Therefore, in the spirit of "definitions are here to be used", most theorems, apart from the most elementary ones, should only have the "br" version, not the "opel" one. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)
Theorem	brabd0 37148*	Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑦𝜒)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))
Theorem	brabd 37149*	Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))
Theorem	bj-brab2a1 37150*	"Unbounded" version of brab2a 5779. (Contributed by BJ, 25-Dec-2023.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓))
21.19.6.2  Identity relation (complements) Complements on the identity relation.
Theorem	bj-opabssvv 37151*	A variant of relopabiv 5830 (which could be proved from it, similarly to relxp 5703 from xpss 5701). (Contributed by BJ, 28-Dec-2023.)
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
Theorem	bj-funidres 37152	The restricted identity relation is a function. (Contributed by BJ, 27-Dec-2023.) TODO: relabel funi 6598 to funid.
⊢ Fun ( I ↾ 𝑉)
Theorem	bj-opelidb 37153	Characterization of the ordered pair elements of the identity relation. Remark: in deduction-style proofs, one could save a few syntactic steps by using another antecedent than ⊤ which already appears in the proof. Here for instance this could be the definition I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} but this would make the proof less easy to read. (Contributed by BJ, 27-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
Theorem	bj-opelidb1 37154	Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 37153 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-inexeqex 37155	Lemma for bj-opelid 37157 (but not specific to the identity relation): if the intersection of two classes is a set and the two classes are equal, then both are sets (all three classes are equal, so they all belong to 𝑉, but it is more convenient to have V in the consequent for theorems using it). (Contributed by BJ, 27-Dec-2023.)
⊢ (((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-elsn0 37156	If the intersection of two classes is a set, then these classes are equal if and only if one is an element of the singleton formed on the other. Stronger form of elsng 4640 and elsn2g 4664 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	bj-opelid 37157	Characterization of the ordered pair elements of the identity relation when the intersection of their components are sets. Note that the antecedent is more general than either component being a set. (Contributed by BJ, 29-Mar-2020.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
Theorem	bj-ideqg 37158	Characterization of the classes related by the identity relation when their intersection is a set. Note that the antecedent is more general than either class being a set. (Contributed by NM, 30-Apr-2004.) Weaken the antecedent to sethood of the intersection. (Revised by BJ, 24-Dec-2023.) TODO: replace ideqg 5862, or at least prove ideqg 5862 from it.
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqgALT 37159	Alternate proof of bj-ideqg 37158 from brabga 5539 instead of bj-opelid 37157 itself proved from bj-opelidb 37153. (Contributed by BJ, 27-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqb 37160	Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-idres 37161	Alternate expression for the restricted identity relation. The advantage of that expression is to expose it as a "bounded" class, being included in the Cartesian square of the restricting class. (Contributed by BJ, 27-Dec-2023.) This is an alternate of idinxpresid 6066 (see idinxpres 6065). See also elrid 6064 and elidinxp 6062. (Proof modification is discouraged.)
⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Theorem	bj-opelidres 37162	Characterization of the ordered pairs in the restricted identity relation when the intersection of their component belongs to the restricting class. TODO: prove bj-idreseq 37163 from it. (Contributed by BJ, 29-Mar-2020.)
⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))
Theorem	bj-idreseq 37163	Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 37158 with V substituted for 𝑉 is a direct consequence of bj-idreseq 37163. This is a strengthening of resieq 6008 which should be proved from it (note that currently, resieq 6008 relies on ideq 5863). Note that the intersection in the antecedent is not very meaningful, but is a device to prove versions with either class assumed to be a set. It could be enough to prove the version with a disjunctive antecedent: ⊢ ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → ...). (Contributed by BJ, 25-Dec-2023.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-idreseqb 37164	Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.)
⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))
Theorem	bj-ideqg1 37165	For sets, the identity relation is the same thing as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalize to a disjunctive antecedent. (Revised by BJ, 24-Dec-2023.) TODO: delete once bj-ideqg 37158 is in the main section.
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqg1ALT 37166	Alternate proof of bj-ideqg1 using brabga 5539 instead of the "unbounded" version bj-brab2a1 37150 or brab2a 5779. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.) TODO: delete once bj-ideqg 37158 is in the main section.
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-opelidb1ALT 37167	Characterization of the couples in I. (Contributed by BJ, 29-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-elid3 37168	Characterization of the couples in I whose first component is a setvar. (Contributed by BJ, 29-Mar-2020.)
⊢ (⟨𝑥, 𝐴⟩ ∈ I ↔ 𝑥 = 𝐴)
Theorem	bj-elid4 37169	Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
Theorem	bj-elid5 37170	Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))
Theorem	bj-elid6 37171	Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐵 ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))
Theorem	bj-elid7 37172	Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
⊢ (⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
21.19.6.3  Functionalized identity (diagonal in a Cartesian square) This subsection defines a functionalized version of the identity relation, that can also be seen as the diagonal in a Cartesian square). As explained in df-bj-diag 37174, it will probably be deleted.
Syntax	cdiag2 37173	Syntax for the diagonal of the Cartesian square of a set.
class Id
Definition	df-bj-diag 37174	Define the functionalized identity, which can also be seen as the diagonal function. Its value is given in bj-diagval 37175 when it is viewed as the functionalized identity, and in bj-diagval2 37176 when it is viewed as the diagonal function. Indeed, Definition df-br 5144 identifies a binary relation with the class of couples that are related by that binary relation (see eqrel2 38300 for the extensionality property of binary relations). As a consequence, the identity relation, or identity function (see funi 6598), on any class, can alternatively be seen as the diagonal of the cartesian square of that class. The identity relation on the universal class, I, is an "identity relation generator", since its restriction to any class is the identity relation on that class. It may be useful to consider a functionalized version of that fact, and that is the purpose of df-bj-diag 37174. Note: most proofs will only use its values (Id‘𝐴), in which case it may be enough to use ( I ↾ 𝐴) everywhere and dispense with this definition. (Contributed by BJ, 22-Jun-2019.)
⊢ Id = (𝑥 ∈ V ↦ ( I ↾ 𝑥))
Theorem	bj-diagval 37175	Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 37176 views it as the diagonal function. See df-bj-diag 37174 for the terminology. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
Theorem	bj-diagval2 37176	Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 37175 views it as the functionalized identity. See df-bj-diag 37174 for the terminology. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Theorem	bj-eldiag 37177	Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 37171. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))
Theorem	bj-eldiag2 37178	Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 37172. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶)))
21.19.6.4  Direct image and inverse image Definitions of the functionalized direct image and inverse image. The functionalized direct (resp. inverse) image is the morphism component of the covariant (resp. contravariant) powerset endofunctor of the category of sets and relations (and, up to restriction, of its subcategory of sets and functions). Its object component is the powerset operation 𝒫 defined in df-pw 4602.
Syntax	cimdir 37179	Syntax for the functionalized direct image.
class 𝒫*
Definition	df-imdir 37180*	Definition of the functionalized direct image, which maps a binary relation between two given sets to its associated direct image relation. (Contributed by BJ, 16-Dec-2023.)
⊢ 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ (𝑟 “ 𝑥) = 𝑦)}))
Theorem	bj-imdirvallem 37181*	Lemma for bj-imdirval 37182 and bj-iminvval 37194. (Contributed by BJ, 23-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝜓)}))    ⇒   ⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)}))
Theorem	bj-imdirval 37182*	Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)}))
Theorem	bj-imdirval2lem 37183*	Lemma for bj-imdirval2 37184 and bj-iminvval2 37195. (Contributed by BJ, 23-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)
Theorem	bj-imdirval2 37184*	Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))    ⇒   ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
Theorem	bj-imdirval3 37185	Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))    ⇒   ⊢ (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))
Theorem	bj-imdiridlem 37186*	Lemma for bj-imdirid 37187 and bj-iminvid 37196. (Contributed by BJ, 26-May-2024.)
⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) → (𝜑 ↔ 𝑥 = 𝑦))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐴) ∧ 𝜑)} = ( I ↾ 𝒫 𝐴)
Theorem	bj-imdirid 37187	Functorial property of the direct image: the direct image by the identity on a set is the identity on the powerset. (Contributed by BJ, 24-Dec-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))
Theorem	bj-opelopabid 37188*	Membership in an ordered-pair class abstraction. One can remove the DV condition on 𝑥, 𝑦 by using opabid 5530 in place of opabidw 5529. (Contributed by BJ, 22-May-2024.)
⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
Theorem	bj-opabco 37189*	Composition of ordered-pair class abstractions. (Contributed by BJ, 22-May-2024.)
⊢ ({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑 ∧ 𝜓)}
Theorem	bj-xpcossxp 37190	The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵 ∩ 𝐶) ≠ ∅, see xpcogend 15013. (Contributed by BJ, 22-May-2024.)
⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)
Theorem	bj-imdirco 37191	Functorial property of the direct image: the direct image by a composition is the composition of the direct images. (Contributed by BJ, 23-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))    &   ⊢ (𝜑 → 𝑆 ⊆ (𝐵 × 𝐶))    ⇒   ⊢ (𝜑 → ((𝐴𝒫*𝐶)‘(𝑆 ∘ 𝑅)) = (((𝐵𝒫*𝐶)‘𝑆) ∘ ((𝐴𝒫*𝐵)‘𝑅)))
Syntax	ciminv 37192	Syntax for the functionalized inverse image.
class 𝒫*
Definition	df-iminv 37193*	Definition of the functionalized inverse image, which maps a binary relation between two given sets to its associated inverse image relation. (Contributed by BJ, 23-Dec-2023.)
⊢ 𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))
Theorem	bj-iminvval 37194*	Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))
Theorem	bj-iminvval2 37195*	Value of the functionalized inverse image. (Contributed by BJ, 23-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))    ⇒   ⊢ (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝑥 = (◡𝑅 “ 𝑦))})
Theorem	bj-iminvid 37196	Functorial property of the inverse image: the inverse image by the identity on a set is the identity on the powerset. (Contributed by BJ, 26-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))
21.19.6.5  Extended numbers and projective lines as sets We parameterize the set of infinite extended complex numbers ℂ∞ (df-bj-ccinfty 37213) using the real numbers ℝ (df-r 11165) via the function +∞eiτ. Since at that point, we have only defined the set of real numbers but no operations on it, we define a temporary "fractional part" function, which is more convenient to define on the temporary reals R (df-nr 11096) since we can use operations on the latter. We also define the temporary real "one-half" in order to define minus infinity (df-bj-minfty 37225) and then we can define the sets of extended real numbers and of extended complex numbers, and the projective real and complex lines, as well as addition and negation on these, and also the order relation on the extended reals (which bypasses the intermediate definition of a temporary order on the real numbers and then a superseding one on the extended real numbers).
Syntax	cfractemp 37197	Syntax for the fractional part of a temporary real.
class {R
Definition	df-bj-fractemp 37198*	Temporary definition: fractional part of a temporary real. To understand this definition, recall the canonical injection ω⟶R, 𝑛 ↦ [{𝑥 ∈ Q ∣ 𝑥 <Q ⟨suc 𝑛, 1o⟩}, 1P] ~R where we successively take the successor of 𝑛 to land in positive integers, then take the couple with 1o as second component to land in positive rationals, then take the Dedekind cut that positive rational forms, and finally take the equivalence class of the couple with 1P as second component. Adding one at the beginning and subtracting it at the end is necessary since the constructions used in set.mm use the positive integers, positive rationals, and positive reals as intermediate number systems. (Contributed by BJ, 22-Jan-2023.) The precise definition is irrelevant and should generally not be used. One could even inline it. The definitive fractional part of an extended or projective complex number will be defined later. (New usage is discouraged.)
⊢ {R = (𝑥 ∈ R ↦ (℩𝑦 ∈ R ((𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))
Syntax	cinftyexpitau 37199	Syntax for the function +∞eiτ parameterizing ℂ∞.
class +∞eiτ
Definition	df-bj-inftyexpitau 37200	Definition of the auxiliary function +∞eiτ parameterizing the circle at infinity ℂ∞ in ℂ̅. We use coupling with {R} to simplify the proof of bj-inftyexpitaudisj 37206. (Contributed by BJ, 22-Jan-2023.) The precise definition is irrelevant and should generally not be used. TODO: prove only the necessary lemmas to prove ⊢ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((+∞eiτ‘𝐴) = (+∞eiτ‘𝐵) ↔ (𝐴 − 𝐵) ∈ ℤ)). (New usage is discouraged.)
⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)