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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-ismoored0 34401 | Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴) | ||
Theorem | bj-ismoored 34402 | 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 34403 | 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 34404* | 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 34405* | Sufficient condition to be a Moore collection (variant of bj-ismooredr 34404 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 34406 | 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 34407 | The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
⊢ ¬ ∅ ∈ Moore | ||
Theorem | bj-snmoore 34408 | A singleton is a Moore collection. See bj-snmooreb 34409 for a biconditional version. (Contributed by BJ, 10-Apr-2024.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Moore) | ||
Theorem | bj-snmooreb 34409 | 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 34410 |
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 34408). 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) | ||
Theorem | bj-0nelmpt 34411 | 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 34412 | 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 34413* | An equivalent definition of df-mpo 7161. (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 34414* | Alternate proof of mpompt 7266. (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 34415 | Syntax for maps-to notation for functions with three arguments. |
class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) | ||
Definition | df-bj-mpt3 34416* | Define maps-to notation for functions with three arguments. See df-mpt 5147 and df-mpo 7161 for functions with one and two arguments respectively. This definition is analogous to bj-dfmpoa 34413. (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)} | ||
Currying and uncurrying. See also df-cur 7933 and df-unc 7934. Contrary to these, the definitions in this section are parameterized. | ||
Syntax | csethom 34417 | Syntax for the set of set morphisms. |
class Set⟶ | ||
Definition | df-bj-sethom 34418* |
Define the set of functions (morphisms of sets) between two sets. Same
as df-map 8408 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 34419 | Syntax for the set of topological morphisms. |
class Top⟶ | ||
Definition | df-bj-tophom 34420* | Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 21835 (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 34421 | Syntax for the set of magma morphisms. |
class Mgm⟶ | ||
Definition | df-bj-mgmhom 34422* | 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 34423 | Syntax for the set of topological magma morphisms. |
class TopMgm⟶ | ||
Definition | df-bj-topmgmhom 34424* | 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- 34425 | Syntax for the parameterized currying function. |
class curry_ | ||
Definition | df-bj-cur 34426* | Define currying. See also df-cur 7933. (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- 34427 | Notation for the parameterized uncurrying function. |
class uncurry_ | ||
Definition | df-bj-unc 34428* | Define uncurrying. See also df-unc 7934. (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)))) | ||
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 34429 | Syntax for component-setting in extensible structures. |
class [𝐵 / 𝐴]struct𝑆 | ||
Definition | df-strset 34430 | 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. (Contributed by BJ, 13-Feb-2022.) |
⊢ [𝐵 / 𝐴]struct𝑆 = ((𝑆 ↾ (V ∖ {(𝐴‘ndx)})) ∪ {〈(𝐴‘ndx), 𝐵〉}) | ||
Theorem | setsstrset 34431 | Relation between df-sets 16490 and df-strset 34430. 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〉)) | ||
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 34518 and df-bj-rrhat 34520, and the point at infinity is denoted by ∞, defined in df-bj-infty 34516. Both ℝ and ℂ also have "directional compactifications", denoted respectively by ℝ̅, defined in df-bj-rrbar 34514 (already defined as ℝ*, see df-xr 10679) and ℂ̅, defined in df-bj-ccbar 34501. 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. | ||
Theorem | bj-nfald 34432 | Variant of nfald 2347. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) | ||
Theorem | bj-nfexd 34433 | Variant of nfexd 2348. (Contributed by BJ, 25-Dec-2023.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) | ||
Theorem | copsex2d 34434* | 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 34435* | Biconditional form of copsex2d 34434. 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 34436* | 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 34437* | 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 34438* | 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 34439 | The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5604, which could be proved from it. (Contributed by BJ, 27-Dec-2023.) |
⊢ ((Rel 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | bj-opelresdm 34440 | If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5859. (Contributed by BJ, 25-Dec-2023.) |
⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋) | ||
Theorem | bj-brresdm 34441 |
If two classes are related by a restricted binary relation, then the first
class is an element of the restricting class. See also brres 5860 and
brrelex1 5605.
Remark: there are many pairs like bj-opelresdm 34440 / bj-brresdm 34441, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 34440 / brrelex12 5604 or the opelopabg 5425 / brabg 5426 family). They are straightforwardly equivalent by df-br 5067. 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 34442* | 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 34443* | 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 34444* | "Unbounded" version of brab2a 5644. (Contributed by BJ, 25-Dec-2023.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓)) | ||
Complements on the identity relation. | ||
Theorem | bj-opabssvv 34445* | A variant of relopabiv 5693 (which could be proved from it, similarly to relxp 5573 from xpss 5571). (Contributed by BJ, 28-Dec-2023.) |
⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ (V × V) | ||
Theorem | bj-funidres 34446 |
The restricted identity relation is a function. (Contributed by BJ,
27-Dec-2023.)
TODO: relabel funi 6387 to funid. |
⊢ Fun ( I ↾ 𝑉) | ||
Theorem | bj-opelidb 34447 |
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 34448 | Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 34447 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 34449 | Lemma for bj-opelid 34451 (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 34450 | 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 4581 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 34451 | 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 34452 |
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 5722, or at least prove ideqg 5722 from it. |
⊢ ((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-ideqgALT 34453 | Alternate proof of bj-ideqg 34452 from brabga 5421 instead of bj-opelid 34451 itself proved from bj-opelidb 34447. (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 34454 | Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.) |
⊢ (𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵)) | ||
Theorem | bj-idres 34455 |
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 5915 (see idinxpres 5914). See also elrid 5913 and elidinxp 5911. (Proof modification is discouraged.) |
⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) | ||
Theorem | bj-opelidres 34456 | 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 34457 from it. (Contributed by BJ, 29-Mar-2020.) |
⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐵〉 ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-idreseq 34457 | Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 34452 with V substituted for 𝑉 is a direct consequence of bj-idreseq 34457. This is a strengthening of resieq 5864 which should be proved from it (note that currently, resieq 5864 relies on ideq 5723). 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 34458 | 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 34459 |
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 34452 is in the main section. |
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-ideqg1ALT 34460 |
Alternate proof of bj-ideqg1 using brabga 5421 instead of the "unbounded"
version bj-brab2a1 34444 or brab2a 5644. (Contributed by BJ, 25-Dec-2023.)
(Proof modification is discouraged.) (New usage is discouraged.)
TODO: delete once bj-ideqg 34452 is in the main section. |
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | bj-opelidb1ALT 34461 | 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 34462 | 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 34463 | Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 ∈ I ↔ (1st ‘𝐴) = (2nd ‘𝐴))) | ||
Theorem | bj-elid5 34464 | Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = (2nd ‘𝐴))) | ||
Theorem | bj-elid6 34465 | 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 34466 | 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)) | ||
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 34468, it will probably be deleted. | ||
Syntax | cdiag2 34467 | Syntax for the diagonal of the Cartesian square of a set. |
class Id | ||
Definition | df-bj-diag 34468 |
Define the functionalized identity, which can also be seen as the diagonal
function. Its value is given in bj-diagval 34469 when it is viewed as the
functionalized identity, and in bj-diagval2 34470 when it is viewed as the
diagonal function.
Indeed, Definition df-br 5067 identifies a binary relation with the class of couples that are related by that binary relation (see eqrel2 35572 for the extensionality property of binary relations). As a consequence, the identity relation, or identity function (see funi 6387), 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 34468. 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 34469 | Value of the funtionalized identity, or equivalently of the diagonal function. This expression views it as the functionalized identity, whereas bj-diagval2 34470 views it as the diagonal function. See df-bj-diag 34468 for the terminology. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ↾ 𝐴)) | ||
Theorem | bj-diagval2 34470 | Value of the funtionalized identity, or equivalently of the diagonal function. This expression views it as the diagonal function, whereas bj-diagval 34469 views it as the functionalized identity. See df-bj-diag 34468 for the terminology. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴))) | ||
Theorem | bj-eldiag 34471 | Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 34465. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))) | ||
Theorem | bj-eldiag2 34472 | Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid7 34466. (Contributed by BJ, 22-Jun-2019.) |
⊢ (𝐴 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (Id‘𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))) | ||
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 4541. | ||
Syntax | cimdir 34473 | Syntax for the functionalized direct image. |
class 𝒫* | ||
Definition | df-imdir 34474* | 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-imdirval 34475* | 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-imdirval2 34476* | 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 34477 | 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-imdirid 34478 | 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)) | ||
Syntax | cinvdir 34479 | Syntax for the functionalized inverse image. |
class 𝒫* | ||
Definition | df-invdir 34480* | 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))})) | ||
We parameterize the set of infinite extended complex numbers ℂ∞ (df-bj-ccinfty 34497) using the real numbers ℝ (df-r 10547) 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 10478) 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 34509) 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 34481 | Syntax for the fractional part of a tempopary real. |
class {R | ||
Definition | df-bj-fractemp 34482* |
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 34483 | Syntax for the function +∞eiτ parameterizing ℂ∞. |
class +∞eiτ | ||
Definition | df-bj-inftyexpitau 34484 | 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 34490. (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 34485 | Syntax for the circle at infinity ℂ∞N. |
class ℂ∞N | ||
Definition | df-bj-ccinftyN 34486 | 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 34487 | The function +∞eiτ written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.) |
⊢ +∞eiτ:ℝ–onto→ℂ∞N | ||
Syntax | chalf 34488 | Syntax for the temporary one-half. |
class 1/2 | ||
Definition | df-bj-onehalf 34489 |
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 7131) $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 34490 | An element of the circle at infinity is not a complex number. (Contributed by BJ, 4-Feb-2023.) |
⊢ ¬ (+∞eiτ‘𝐴) ∈ ℂ | ||
Syntax | cinftyexpi 34491 | Syntax for the function +∞ei parameterizing ℂ∞. |
class +∞ei | ||
Definition | df-bj-inftyexpi 34492 | 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 34498. 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 34484 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 34493 | 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 34494 | Injectivity of the parameterization +∞ei. Remark: a more conceptual proof would use bj-inftyexpiinv 34493 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 34495 | 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 34496 | Syntax for the circle at infinity ℂ∞. |
class ℂ∞ | ||
Definition | df-bj-ccinfty 34497 | 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 34498 | The circle at infinity is disjoint from the set of complex numbers. (Contributed by BJ, 22-Jun-2019.) |
⊢ (ℂ ∩ ℂ∞) = ∅ | ||
Theorem | bj-elccinfty 34499 | A lemma for infinite extended complex numbers. (Contributed by BJ, 27-Jun-2019.) |
⊢ (𝐴 ∈ (-π(,]π) → (+∞ei‘𝐴) ∈ ℂ∞) | ||
Syntax | cccbar 34500 | Syntax for the set of extended complex numbers ℂ̅. |
class ℂ̅ |
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