P. 374 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-bj-tophom 37301*	Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 23175 (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 37302	Syntax for the set of magma morphisms.
class Mgm⟶
Definition	df-bj-mgmhom 37303*	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 37304	Syntax for the set of topological magma morphisms.
class TopMgm⟶
Definition	df-bj-topmgmhom 37305*	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- 37306	Syntax for the parameterized currying function.
class curry_
Definition	df-bj-cur 37307*	Define currying. See also df-cur 8211. (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- 37308	Notation for the parameterized uncurrying function.
class uncurry_
Definition	df-bj-unc 37309*	Define uncurrying. See also df-unc 8212. (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 37310	Syntax for component-setting in extensible structures.
class [𝐵 / 𝐴]struct𝑆
Definition	df-strset 37311	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 17095, but since extensible structures are functions on ℕ, it will be more natural to replace it with ℕ when df-strset 37311 becomes the main definition. (Contributed by BJ, 13-Feb-2022.)
⊢ [𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {⟨(𝐴‘ndx), 𝐵⟩})
Theorem	setsstrset 37312	Relation between df-sets 17095 and df-strset 37311. 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 37409 and df-bj-rrhat 37411, and the point at infinity is denoted by ∞, defined in df-bj-infty 37407. Both ℝ and ℂ also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 37405 (already defined as ℝ*, see df-xr 11174) and ℂ̅, defined in df-bj-ccbar 37392. 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 37313	Variant of nfald 2334. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	bj-nfexd 37314	Variant of nfexd 2335. (Contributed by BJ, 25-Dec-2023.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
Theorem	copsex2d 37315*	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 37316*	Biconditional form of copsex2d 37315. 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 37317*	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 37318*	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 37319*	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 37320	The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5677, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)
⊢ ((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	bj-opelresdm 37321	If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5945. (Contributed by BJ, 25-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
Theorem	bj-brresdm 37322	If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 5946 and brrelex1 5678. Remark: there are many pairs like bj-opelresdm 37321 / bj-brresdm 37322, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 37321 / brrelex12 5677 or the opelopabg 5487 / brabg 5488 family). They are straightforwardly equivalent by df-br 5100. 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 37323*	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 37324*	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 37325*	"Unbounded" version of brab2a 5718. (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 37326*	A variant of relopabiv 5770 (which could be proved from it, similarly to relxp 5643 from xpss 5641). (Contributed by BJ, 28-Dec-2023.)
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
Theorem	bj-funidres 37327	The restricted identity relation is a function. (Contributed by BJ, 27-Dec-2023.) TODO: relabel funi 6525 to funid.
⊢ Fun ( I ↾ 𝑉)
Theorem	bj-opelidb 37328	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 37329	Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 37328 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 37330	Lemma for bj-opelid 37332 (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 37331	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 4595 and elsn2g 4622 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	bj-opelid 37332	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 37333	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 5801, or at least prove ideqg 5801 from it.
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqgALT 37334	Alternate proof of bj-ideqg 37333 from brabga 5483 instead of bj-opelid 37332 itself proved from bj-opelidb 37328. (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 37335	Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-idres 37336	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 6008 (see idinxpres 6007). See also elrid 6006 and elidinxp 6004. (Proof modification is discouraged.)
⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Theorem	bj-opelidres 37337	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 37338 from it. (Contributed by BJ, 29-Mar-2020.)
⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))
Theorem	bj-idreseq 37338	Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 37333 with V substituted for 𝑉 is a direct consequence of bj-idreseq 37338. This is a strengthening of resieq 5950 which should be proved from it (note that currently, resieq 5950 relies on ideq 5802). 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 37339	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 37340	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 37333 is in the main section.
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqg1ALT 37341	Alternate proof of bj-ideqg1 using brabga 5483 instead of the "unbounded" version bj-brab2a1 37325 or brab2a 5718. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.) TODO: delete once bj-ideqg 37333 is in the main section.
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-opelidb1ALT 37342	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 37343	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 37344	Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
Theorem	bj-elid5 37345	Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))
Theorem	bj-elid6 37346	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 37347	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 37349, it will probably be deleted.
Syntax	cdiag2 37348	Syntax for the diagonal of the Cartesian square of a set.
class Id
Definition	df-bj-diag 37349	Define the functionalized identity, which can also be seen as the diagonal function. Its value is given in bj-diagval 37350 when it is viewed as the functionalized identity, and in bj-diagval2 37351 when it is viewed as the diagonal function. Indeed, Definition df-br 5100 identifies a binary relation with the class of couples that are related by that binary relation (see eqrel2 38477 for the extensionality property of binary relations). As a consequence, the identity relation, or identity function (see funi 6525), 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 37349. 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 37350	Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 37351 views it as the diagonal function. See df-bj-diag 37349 for the terminology. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
Theorem	bj-diagval2 37351	Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 37350 views it as the functionalized identity. See df-bj-diag 37349 for the terminology. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Theorem	bj-eldiag 37352	Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 37346. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))
Theorem	bj-eldiag2 37353	Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 37347. (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 4557.
Syntax	cimdir 37354	Syntax for the functionalized direct image.
class 𝒫*
Definition	df-imdir 37355*	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 37356*	Lemma for bj-imdirval 37357 and bj-iminvval 37369. (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 37357*	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 37358*	Lemma for bj-imdirval2 37359 and bj-iminvval2 37370. (Contributed by BJ, 23-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)
Theorem	bj-imdirval2 37359*	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 37360	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 37361*	Lemma for bj-imdirid 37362 and bj-iminvid 37371. (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 37362	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 37363*	Membership in an ordered-pair class abstraction. One can remove the DV condition on 𝑥, 𝑦 by using opabid 5474 in place of opabidw 5473. (Contributed by BJ, 22-May-2024.)
⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
Theorem	bj-opabco 37364*	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 37365	The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵 ∩ 𝐶) ≠ ∅, see xpcogend 14901. (Contributed by BJ, 22-May-2024.)
⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)
Theorem	bj-imdirco 37366	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 37367	Syntax for the functionalized inverse image.
class 𝒫*
Definition	df-iminv 37368*	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 37369*	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 37370*	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 37371	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 37388) using the real numbers ℝ (df-r 11040) 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 10971) 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 37400) 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 37372	Syntax for the fractional part of a temporary real.
class {R
Definition	df-bj-fractemp 37373*	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 37374	Syntax for the function +∞eiτ parameterizing ℂ∞.
class +∞eiτ
Definition	df-bj-inftyexpitau 37375	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 37381. (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}⟩)
Syntax	cccinftyN 37376	Syntax for the circle at infinity ℂ∞N.
class ℂ∞N
Definition	df-bj-ccinftyN 37377	Definition of the circle at infinity ℂ∞N. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
⊢ ℂ∞N = ran +∞eiτ
Theorem	bj-inftyexpitaufo 37378	The function +∞eiτ written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.)
⊢ +∞eiτ:ℝ–onto→ℂ∞N
Syntax	chalf 37379	Syntax for the temporary one-half.
class 1/2
Definition	df-bj-onehalf 37380	Define the temporary real "one-half". Once the machinery is developed, the real number "one-half" is commonly denoted by (1 / 2). (Contributed by BJ, 4-Feb-2023.) (New usage is discouraged.) TODO: $p |- 1/2 e. R. $= ? $. (riotacl 7334) $p |- -. 0R = 1/2 $= ? $. (since -. ( 0R +R 0R ) = 1R ) $p |- 0R <R 1/2 $= ? $. $p |- 1/2 <R 1R $= ? $. $p |- ( {R ` 0R ) = 0R $= ? $. $p |- ( {R ` 1/2 ) = 1/2 $= ? $. df-minfty $a |- minfty = ( inftyexpitau ` <. 1/2 , 0R >. ) $.
⊢ 1/2 = (℩𝑥 ∈ R (𝑥 +R 𝑥) = 1R)
Theorem	bj-inftyexpitaudisj 37381	An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.)
⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ
Syntax	cinftyexpi 37382	Syntax for the function +∞ei parameterizing ℂ∞.
class +∞ei
Definition	df-bj-inftyexpi 37383	Definition of the auxiliary function +∞ei parameterizing the circle at infinity ℂ∞ in ℂ̅. We use coupling with ℂ to simplify the proof of bj-ccinftydisj 37389. It could seem more natural to define +∞ei on all of ℝ, but we want to use only basic functions in the definition of ℂ̅. TODO: transition to df-bj-inftyexpitau 37375 instead. (Contributed by BJ, 22-Jun-2019.) The precise definition is irrelevant and should generally not be used. (New usage is discouraged.)
⊢ +∞ei = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
Theorem	bj-inftyexpiinv 37384	Utility theorem for the inverse of +∞ei. (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 ‘(+∞ei‘𝐴)) = 𝐴)
Theorem	bj-inftyexpiinj 37385	Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 37384 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 ↔ (+∞ei‘𝐴) = (+∞ei‘𝐵)))
Theorem	bj-inftyexpidisj 37386	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.)
⊢ ¬ (+∞ei‘𝐴) ∈ ℂ
Syntax	cccinfty 37387	Syntax for the circle at infinity ℂ∞.
class ℂ∞
Definition	df-bj-ccinfty 37388	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 +∞ei
Theorem	bj-ccinftydisj 37389	The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
⊢ (ℂ ∩ ℂ∞) = ∅
Theorem	bj-elccinfty 37390	A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) ∈ ℂ∞)
Syntax	cccbar 37391	Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅
Definition	df-bj-ccbar 37392	Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
⊢ ℂ̅ = (ℂ ∪ ℂ∞)
Theorem	bj-ccssccbar 37393	Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ ⊆ ℂ̅
Theorem	bj-ccinftyssccbar 37394	Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ∞ ⊆ ℂ̅
Syntax	cpinfty 37395	Syntax for "plus infinity".
class +∞
Definition	df-bj-pinfty 37396	Definition of "plus infinity". (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ = (+∞ei‘0)
Theorem	bj-pinftyccb 37397	The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ∈ ℂ̅
Theorem	bj-pinftynrr 37398	The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ +∞ ∈ ℂ
Syntax	cminfty 37399	Syntax for "minus infinity".
class -∞
Definition	df-bj-minfty 37400	Definition of "minus infinity". (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ = (+∞ei‘π)