P. 376 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37501-37600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	copsex2b 37501*	Biconditional form of copsex2d 37500. 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 37502*	Membership of an ordered 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 37503*	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 37504*	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 37505	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 37506	If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5944. (Contributed by BJ, 25-Dec-2023.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
Theorem	bj-brresdm 37507	If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 5945 and brrelex1 5678. Remark: there are many pairs like bj-opelresdm 37506 / bj-brresdm 37507, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 37506 / brrelex12 5677 or the opelopabg 5487 / brabg 5488 family). They are straightforwardly equivalent by df-br 5080. 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 37508*	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 37509*	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 37510*	"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 37511*	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 37512	The restricted identity relation is a function. (Contributed by BJ, 27-Dec-2023.) TODO: relabel funi 6524 to funid.
⊢ Fun ( I ↾ 𝑉)
Theorem	bj-opelidb 37513	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 37514	Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 37513 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 37515	Lemma for bj-opelid 37517 (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 37516	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 4576 and elsn2g 4603 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
Theorem	bj-opelid 37517	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 37518	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 5800, or at least prove ideqg 5800 from it.
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqgALT 37519	Alternate proof of bj-ideqg 37518 from brabga 5483 instead of bj-opelid 37517 itself proved from bj-opelidb 37513. (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 37520	Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Theorem	bj-idres 37521	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 6007 (see idinxpres 6006). See also elrid 6005 and elidinxp 6003. (Proof modification is discouraged.)
⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Theorem	bj-opelidres 37522	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 37523 from it. (Contributed by BJ, 29-Mar-2020.)
⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))
Theorem	bj-idreseq 37523	Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 37518 with V substituted for 𝑉 is a direct consequence of bj-idreseq 37523. This is a strengthening of resieq 5949 which should be proved from it (note that currently, resieq 5949 relies on ideq 5801). 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 37524	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 37525	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 37518 is in the main section.
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-ideqg1ALT 37526	Alternate proof of bj-ideqg1 using brabga 5483 instead of the "unbounded" version bj-brab2a1 37510 or brab2a 5718. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.) TODO: delete once bj-ideqg 37518 is in the main section.
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	bj-opelidb1ALT 37527	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 37528	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 37529	Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴)))
Theorem	bj-elid5 37530	Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴)))
Theorem	bj-elid6 37531	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 37532	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 37534, it will probably be deleted.
Syntax	cdiag2 37533	Syntax for the diagonal of the Cartesian square of a set.
class Id
Definition	df-bj-diag 37534	Define the functionalized identity, which can also be seen as the diagonal function. Its value is given in bj-diagval 37535 when it is viewed as the functionalized identity, and in bj-diagval2 37536 when it is viewed as the diagonal function. Indeed, Definition df-br 5080 identifies a binary relation with the class of couples that are related by that binary relation (see eqrel2 38673 for the extensionality property of binary relations). As a consequence, the identity relation, or identity function (see funi 6524), 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 37534. 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 37535	Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 37536 views it as the diagonal function. See df-bj-diag 37534 for the terminology. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴))
Theorem	bj-diagval2 37536	Value of the functionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 37535 views it as the functionalized identity. See df-bj-diag 37534 for the terminology. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
Theorem	bj-eldiag 37537	Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 37531. (Contributed by BJ, 22-Jun-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))
Theorem	bj-eldiag2 37538	Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 37532. (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 4538.
Syntax	cimdir 37539	Syntax for the functionalized direct image.
class 𝒫*
Definition	df-imdir 37540*	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 37541*	Lemma for bj-imdirval 37542 and bj-iminvval 37554. (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 37542*	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 37543*	Lemma for bj-imdirval2 37544 and bj-iminvval2 37555. (Contributed by BJ, 23-May-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ 𝜓)} ∈ V)
Theorem	bj-imdirval2 37544*	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 37545	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 37546*	Lemma for bj-imdirid 37547 and bj-iminvid 37556. (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 37547	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 37548*	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 37549*	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 37550	The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵 ∩ 𝐶) ≠ ∅, see xpcogend 14934. (Contributed by BJ, 22-May-2024.)
⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷)
Theorem	bj-imdirco 37551	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 37552	Syntax for the functionalized inverse image.
class 𝒫*
Definition	df-iminv 37553*	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 37554*	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 37555*	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 37556	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 37573) using the real numbers ℝ (df-r 11046) 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 10977) 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 37585) 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 37557	Syntax for the fractional part of a temporary real.
class {R
Definition	df-bj-fractemp 37558*	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 37559	Syntax for the function +∞eiτ parameterizing ℂ∞.
class +∞eiτ
Definition	df-bj-inftyexpitau 37560	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 37566. (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 37561	Syntax for the circle at infinity ℂ∞N.
class ℂ∞N
Definition	df-bj-ccinftyN 37562	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 37563	The function +∞eiτ written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.)
⊢ +∞eiτ:ℝ–onto→ℂ∞N
Syntax	chalf 37564	Syntax for the temporary one-half.
class 1/2
Definition	df-bj-onehalf 37565	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 7337) $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 37566	An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.)
⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ
Syntax	cinftyexpi 37567	Syntax for the function +∞ei parameterizing ℂ∞.
class +∞ei
Definition	df-bj-inftyexpi 37568	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 37574. 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 37560 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 37569	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 37570	Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 37569 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 37571	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 37572	Syntax for the circle at infinity ℂ∞.
class ℂ∞
Definition	df-bj-ccinfty 37573	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 37574	The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.)
⊢ (ℂ ∩ ℂ∞) = ∅
Theorem	bj-elccinfty 37575	A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) ∈ ℂ∞)
Syntax	cccbar 37576	Syntax for the set of extended complex numbers ℂ̅.
class ℂ̅
Definition	df-bj-ccbar 37577	Definition of the set of extended complex numbers ℂ̅. (Contributed by BJ, 22-Jun-2019.)
⊢ ℂ̅ = (ℂ ∪ ℂ∞)
Theorem	bj-ccssccbar 37578	Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ ⊆ ℂ̅
Theorem	bj-ccinftyssccbar 37579	Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ∞ ⊆ ℂ̅
Syntax	cpinfty 37580	Syntax for "plus infinity".
class +∞
Definition	df-bj-pinfty 37581	Definition of "plus infinity". (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ = (+∞ei‘0)
Theorem	bj-pinftyccb 37582	The class +∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ∈ ℂ̅
Theorem	bj-pinftynrr 37583	The extended complex number +∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ +∞ ∈ ℂ
Syntax	cminfty 37584	Syntax for "minus infinity".
class -∞
Definition	df-bj-minfty 37585	Definition of "minus infinity". (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ = (+∞ei‘π)
Theorem	bj-minftyccb 37586	The class -∞ is an extended complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ -∞ ∈ ℂ̅
Theorem	bj-minftynrr 37587	The extended complex number -∞ is not a complex number. (Contributed by BJ, 27-Jun-2019.)
⊢ ¬ -∞ ∈ ℂ
Theorem	bj-pinftynminfty 37588	The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.)
⊢ +∞ ≠ -∞
Syntax	crrbar 37589	Syntax for the set of extended real numbers.
class ℝ̅
Definition	df-bj-rrbar 37590	Definition of the set of extended real numbers. This aims to replace df-xr 11181. (Contributed by BJ, 29-Jun-2019.)
⊢ ℝ̅ = (ℝ ∪ {-∞, +∞})
Syntax	cinfty 37591	Syntax for ∞.
class ∞
Definition	df-bj-infty 37592	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 37593	Syntax for ℂ̂.
class ℂ̂
Definition	df-bj-cchat 37594	Define the complex projective line, or Riemann sphere. (Contributed by BJ, 27-Jun-2019.)
⊢ ℂ̂ = (ℂ ∪ {∞})
Syntax	crrhat 37595	Syntax for ℝ̂.
class ℝ̂
Definition	df-bj-rrhat 37596	Define the real projective line. (Contributed by BJ, 27-Jun-2019.)
⊢ ℝ̂ = (ℝ ∪ {∞})
Theorem	bj-rrhatsscchat 37597	The real projective line is included in the complex projective line. (Contributed by BJ, 27-Jun-2019.)
⊢ ℝ̂ ⊆ ℂ̂
21.19.6.6  Addition and opposite We define the operations of addition and opposite on the extended complex numbers and on the complex projective line (Riemann sphere) simultaneously, thus "overloading" the operations.
Syntax	caddcc 37598	Syntax for the addition on extended complex numbers.
class +ℂ̅
Definition	df-bj-addc 37599	Define the additions on the extended complex numbers (on the subset of (ℂ̅ × ℂ̅) where it makes sense) and on the complex projective line (Riemann sphere). We use the plural in "additions" since these are two different operations, even though +ℂ̅ is overloaded. (Contributed by BJ, 22-Jun-2019.)
⊢ +ℂ̅ = (𝑥 ∈ (((ℂ × ℂ̅) ∪ (ℂ̅ × ℂ)) ∪ ((ℂ̂ × ℂ̂) ∪ ( I ↾ ℂ∞))) ↦ if(((1st ‘𝑥) = ∞ ∨ (2nd ‘𝑥) = ∞), ∞, if((1st ‘𝑥) ∈ ℂ, if((2nd ‘𝑥) ∈ ℂ, ⟨((1st ‘(1st ‘𝑥)) +R (1st ‘(2nd ‘𝑥))), ((2nd ‘(1st ‘𝑥)) +R (2nd ‘(2nd ‘𝑥)))⟩, (2nd ‘𝑥)), (1st ‘𝑥))))
Syntax	coppcc 37600	Syntax for negation on the set of extended complex numbers and the complex projective line (Riemann sphere).
class -ℂ̅