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Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcotrintab 40401 The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.)
(𝜑 → (𝑥𝑥) ⊆ 𝑥)       ( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑}

Theoremrclexi 40402* The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
𝐴𝑉        {𝑥 ∣ (𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V

Theoremrtrclexlem 40403 Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.)
(𝑅𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)

Theoremrtrclex 40404* The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.)
(𝐴 ∈ V ↔ {𝑥 ∣ (𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)

TheoremtrclubgNEW 40405* If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

TheoremtrclubNEW 40406* If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑 → Rel 𝑅)       (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))

Theoremtrclexi 40407* The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
𝐴𝑉        {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V

Theoremrtrclexi 40408* The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
𝐴𝑉        {𝑥 ∣ (𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V

Theoremclrellem 40409* When the property 𝜓 holds for a relation substituted for 𝑥, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.)
(𝜑𝑌 ∈ V)    &   (𝜑 → Rel 𝑋)    &   (𝑥 = 𝑌 → (𝜓𝜒))    &   (𝜑𝑋𝑌)    &   (𝜑𝜒)       (𝜑 → Rel {𝑥 ∣ (𝑋𝑥𝜓)})

Theoremclcnvlem 40410* When 𝐴, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
((𝜑𝑥 = (𝑦 ∪ (𝑋𝑋))) → (𝜒𝜓))    &   ((𝜑𝑦 = 𝑥) → (𝜓𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝜑𝑋𝐴)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝜃)       (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} = {𝑦 ∣ (𝑋𝑦𝜒)})

Theoremcnvtrucl0 40411* The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
(𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ⊤)} = {𝑦 ∣ (𝑋𝑦 ∧ ⊤)})

Theoremcnvrcl0 40412* The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
(𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = {𝑦 ∣ (𝑋𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})

Theoremcnvtrcl0 40413* The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
(𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑋𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)})

Theoremdmtrcl 40414* The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.)
(𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = dom 𝑋)

Theoremrntrcl 40415* The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)
(𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = ran 𝑋)

Theoremdfrtrcl5 40416* Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)
t* = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))})

20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures

Theoremtrcleq2lemRP 40417 Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

20.31.1.17  Additions for square root; absolute value

This is based on the observation that the real and imaginary parts of a complex number can be calculated from the number's absolute and real part and the sign of its imaginary part. Formalization of the formula in sqrtcval 40428 was motivated by a short Michael Penn video.

Theoremsqrtcvallem1 40418 Two ways of saying a complex number does not lie on the positive real axis. Lemma for sqrtcval 40428. (Contributed by RP, 17-May-2024.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (((ℑ‘𝐴) = 0 → (ℜ‘𝐴) ≤ 0) ↔ ¬ 𝐴 ∈ ℝ+))

Theoremreabsifneg 40419 Alternate expression for the absolute value of a real number. Lemma for sqrtcval 40428. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 < 0, -𝐴, 𝐴))

Theoremreabsifnpos 40420 Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 ≤ 0, -𝐴, 𝐴))

Theoremreabsifpos 40421 Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℝ → (abs‘𝐴) = if(0 < 𝐴, 𝐴, -𝐴))

Theoremreabsifnneg 40422 Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℝ → (abs‘𝐴) = if(0 ≤ 𝐴, 𝐴, -𝐴))

Theoremreabssgn 40423 Alternate expression for the absolute value of a real number. (Contributed by RP, 22-May-2024.)
(𝐴 ∈ ℝ → (abs‘𝐴) = ((sgn‘𝐴) · 𝐴))

Theoremsqrtcvallem2 40424 Equivalent to saying that the square of the imaginary component of the square root of a complex number is a non-negative real number. Lemma for sqrtcval 40428. See imsqrtval 40431. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) − (ℜ‘𝐴)) / 2))

Theoremsqrtcvallem3 40425 Equivalent to saying that the absolute value of the imaginary component of the square root of a complex number is a real number. Lemma for sqrtcval 40428, sqrtcval2 40429, resqrtval 40430, and imsqrtval 40431. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℂ → (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)) ∈ ℝ)

Theoremsqrtcvallem4 40426 Equivalent to saying that the square of the real component of the square root of a complex number is a non-negative real number. Lemma for sqrtcval 40428. See resqrtval 40430. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2))

Theoremsqrtcvallem5 40427 Equivalent to saying that the real component of the square root of a complex number is a real number. Lemma for resqrtval 40430 and imsqrtval 40431. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℂ → (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) ∈ ℝ)

Theoremsqrtcval 40428 Explicit formula for the complex square root in terms of the square root of non-negative reals. The right-hand side is decomposed into real and imaginary parts in the format expected by crrei 14560 and crimi 14561. This formula can be found in section 3.7.27 of Handbook of Mathematical Functions, ed. M. Abramowitz and I. A. Stegun (1965, Dover Press). (Contributed by RP, 18-May-2024.)
(𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (i · (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))))

Theoremsqrtcval2 40429 Explicit formula for the complex square root in terms of the square root of non-negative reals. The right side is slightly more compact than sqrtcval 40428. (Contributed by RP, 18-May-2024.)
(𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (if((ℑ‘𝐴) < 0, -i, i) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)))))

Theoremresqrtval 40430 Real part of the complex square root. (Contributed by RP, 18-May-2024.)
(𝐴 ∈ ℂ → (ℜ‘(√‘𝐴)) = (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)))

Theoremimsqrtval 40431 Imaginary part of the complex square root. (Contributed by RP, 18-May-2024.)
(𝐴 ∈ ℂ → (ℑ‘(√‘𝐴)) = (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))

Theoremresqrtvalex 40432 Example for resqrtval 40430. (Contributed by RP, 21-May-2024.)
(ℜ‘(√‘(15 + (i · 8)))) = 4

Theoremimsqrtvalex 40433 Example for imsqrtval 40431. (Contributed by RP, 21-May-2024.)
(ℑ‘(√‘(15 + (i · 8)))) = 1

20.31.2  Additional statements on relations and subclasses

Theoremal3im 40434 Version of ax-4 1811 for a nested implication. (Contributed by RP, 13-Apr-2020.)
(∀𝑥(𝜑 → (𝜓 → (𝜒𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))

Theoremintima0 40435* Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
𝑎𝐴 (𝑎𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}

Theoremelimaint 40436* Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
(𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)

Theoremcsbcog 40437 Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremcnviun 40438* Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Theoremimaiun1 40439* The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremcoiun1 40440* Composition with an indexed union. Proof analgous to that of coiun 6081. (Contributed by RP, 20-Jun-2020.)
( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)

Theoremelintima 40441* Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
(𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)

Theoremintimass 40442* The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
( 𝐴𝐵) ⊆ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}

Theoremintimass2 40443* The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
( 𝐴𝐵) ⊆ 𝑥𝐴 (𝑥𝐵)

Theoremintimag 40444* Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.)
(∀𝑦(∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ( 𝐴𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)})

Theoremintimasn 40445* Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
(𝐵𝑉 → ( 𝐴 “ {𝐵}) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎 “ {𝐵})})

Theoremintimasn2 40446* Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
(𝐵𝑉 → ( 𝐴 “ {𝐵}) = 𝑥𝐴 (𝑥 “ {𝐵}))

Theoremss2iundf 40447* Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
𝑥𝜑    &   𝑦𝜑    &   𝑦𝑌    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   𝑦𝐶    &   𝑥𝐷    &   𝑦𝐺    &   ((𝜑𝑥𝐴) → 𝑌𝐶)    &   ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ((𝜑𝑥𝐴) → 𝐵𝐺)       (𝜑 𝑥𝐴 𝐵 𝑦𝐶 𝐷)

Theoremss2iundv 40448* Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
((𝜑𝑥𝐴) → 𝑌𝐶)    &   ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ((𝜑𝑥𝐴) → 𝐵𝐺)       (𝜑 𝑥𝐴 𝐵 𝑦𝐶 𝐷)

Theoremcbviuneq12df 40449* Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝑋    &   𝑦𝑌    &   𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   𝑦𝐶    &   𝑥𝐷    &   𝑥𝐹    &   𝑦𝐺    &   ((𝜑𝑦𝐶) → 𝑋𝐴)    &   ((𝜑𝑥𝐴) → 𝑌𝐶)    &   ((𝜑𝑦𝐶𝑥 = 𝑋) → 𝐵 = 𝐹)    &   ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐺)    &   ((𝜑𝑦𝐶) → 𝐷 = 𝐹)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)

Theoremcbviuneq12dv 40450* Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
((𝜑𝑦𝐶) → 𝑋𝐴)    &   ((𝜑𝑥𝐴) → 𝑌𝐶)    &   ((𝜑𝑦𝐶𝑥 = 𝑋) → 𝐵 = 𝐹)    &   ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐺)    &   ((𝜑𝑦𝐶) → 𝐷 = 𝐹)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)

Theoremconrel1d 40451 Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐴𝐵) = ∅)

Theoremconrel2d 40452 Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐵𝐴) = ∅)

20.31.2.1  Transitive relations (not to be confused with transitive classes).

Theoremtrrelind 40453 The intersection of transitive relations is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑 → (𝑆𝑆) ⊆ 𝑆)    &   (𝜑𝑇 = (𝑅𝑆))       (𝜑 → (𝑇𝑇) ⊆ 𝑇)

Theoremxpintrreld 40454 The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))       (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Theoremrestrreld 40455 The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆 = (𝑅𝐴))       (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperreldg 40456 Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 25-Dec-2019.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑆 = (dom 𝑅 × ran 𝑅))       (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))

Theoremtrficl 40457* The class of all transitive relations has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)
𝐴 = {𝑧 ∣ (𝑧𝑧) ⊆ 𝑧}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremcnvtrrel 40458 The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)
((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperrel2dg 40459 Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 20-Jul-2020.)
(𝜑𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))       (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))

20.31.2.2  Reflexive closures

Syntaxcrcl 40460 Extend class notation with reflexive closure.
class r*

Definitiondf-rcl 40461* Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.)
r* = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})

Theoremdfrcl2 40462 Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))

Theoremdfrcl3 40463 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ ((𝑥𝑟0) ∪ (𝑥𝑟1)))

Theoremdfrcl4 40464* Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.)
r* = (𝑟 ∈ V ↦ 𝑛 ∈ {0, 1} (𝑟𝑟𝑛))

20.31.2.3  Finite relationship composition.

In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of 𝑟 that works on finite ordinals or finite sets instead of natural numbers.

Theoremrelexp2 40465 A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.)
(𝑅𝑉 → (𝑅𝑟2) = (𝑅𝑅))

Theoremrelexpnul 40466 If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.)
(((𝑅𝑉 ∧ Rel 𝑅) ∧ (𝑁 ∈ ℕ0𝑀 ∈ ℕ0)) → ((dom (𝑅𝑟𝑁) ∩ ran (𝑅𝑟𝑀)) = ∅ ↔ (𝑅𝑟(𝑁 + 𝑀)) = ∅))

Theoremeliunov2 40467* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. Generalized from dfrtrclrec2 14426. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

Theoremeltrclrec 40468* Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅𝑟𝑛)))

Theoremelrtrclrec 40469* Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅𝑟𝑛)))

Theorembriunov2 40470* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))

Theorembrmptiunrelexpd 40471* If two elements are connected by an indexed union of relational powers, then they are connected via 𝑛 instances the relation, for some 𝑛. Generalization of dfrtrclrec2 14426. (Contributed by RP, 21-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ⊆ ℕ0)       (𝜑 → (𝐴(𝐶𝑅)𝐵 ↔ ∃𝑛𝑁 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvmptiunrelexplb0d 40472* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))

Theoremfvmptiunrelexplb0da 40473* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → Rel 𝑅)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))

Theoremfvmptiunrelexplb1d 40474* If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 1 ∈ 𝑁)       (𝜑𝑅 ⊆ (𝐶𝑅))

Theorembrfvid 40475 If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

TheorembrfvidRP 40476 If two elements are connected by a value of the identity relation, then they are connected via the argument. This is an example which uses brmptiunrelexpd 40471. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

Theoremfvilbd 40477 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))

TheoremfvilbdRP 40478 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))

Theorembrfvrcld 40479 If two elements are connected by the reflexive closure of a relation, then they are connected via zero or one instances the relation. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅𝑟0)𝐵𝐴(𝑅𝑟1)𝐵)))

Theorembrfvrcld2 40480 If two elements are connected by the reflexive closure of a relation, then they are equal or related by relation. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))

Theoremfvrcllb0d 40481 A restriction of the identity relation is a subset of the reflexive closure of a set. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (r*‘𝑅))

Theoremfvrcllb0da 40482 A restriction of the identity relation is a subset of the reflexive closure of a relation. (Contributed by RP, 22-Jul-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (r*‘𝑅))

Theoremfvrcllb1d 40483 A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (r*‘𝑅))

Theorembrtrclrec 40484* Two classes related by the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ 𝑋(𝑅𝑟𝑛)𝑌))

Theorembrrtrclrec 40485* Two classes related by the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅𝑟𝑛)𝑌))

Theorembriunov2uz 40486* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))

Theoremeliunov2uz 40487* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

Theoremov2ssiunov2 40488* Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14425 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))

Theoremrelexp0eq 40489 The zeroth power of relationships is the same if and only if the union of their domain and ranges is the same. (Contributed by RP, 11-Jun-2020.)
((𝐴𝑈𝐵𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴𝑟0) = (𝐵𝑟0)))

Theoremiunrelexp0 40490* Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))

Theoremrelexpxpnnidm 40491 Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
(𝑁 ∈ ℕ → ((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑𝑟𝑁) = (𝐴 × 𝐵)))

Theoremrelexpiidm 40492 Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → (( I ↾ 𝐴)↑𝑟𝑁) = ( I ↾ 𝐴))

Theoremrelexpss1d 40493 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))

Theoremcomptiunov2i 40494* The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)
𝑋 = (𝑎 ∈ V ↦ 𝑖𝐼 (𝑎 𝑖))    &   𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))    &   𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))    &   𝐼 ∈ V    &   𝐽 ∈ V    &   𝐾 = (𝐼𝐽)    &    𝑘𝐼 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)    &    𝑘𝐽 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)    &    𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ⊆ 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)       (𝑋𝑌) = 𝑍

Theoremcorclrcl 40495 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(r* ∘ r*) = r*

Theoremiunrelexpmin1 40496* The indexed union of relation exponentiation over the natural numbers is the minimum transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = ℕ) → ∀𝑠((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))

Theoremrelexpmulnn 40497 With exponents limited to the counting numbers, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
(((𝑅𝑉𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Theoremrelexpmulg 40498 With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
(((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) ∧ (𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Theoremtrclrelexplem 40499* The union of relational powers to positive multiples of 𝑁 is a subset to the transitive closure raised to the power of 𝑁. (Contributed by RP, 15-Jun-2020.)
(𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))

Theoremiunrelexpmin2 40500* The indexed union of relation exponentiation over the natural numbers (including zero) is the minimum reflexive-transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45415
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