Home Metamath Proof ExplorerTheorem List (p. 393 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 39201-39300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrnmptssd 39201* The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → ran 𝐹𝐶)

Theoremprojf1o 39202* A biijection from a set to a projection in a two dimensional space. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   𝐹 = (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩)       (𝜑𝐹:𝐵1-1-onto→({𝐴} × 𝐵))

Theoremfvmap 39203 Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 ∈ (𝐴𝑚 𝐵))    &   (𝜑𝐶𝐵)       (𝜑 → (𝐹𝐶) ∈ 𝐴)

Theoremmapsnd 39204* The value of set exponentiation with a singleton exponent. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})

Theoremfvixp2 39205* Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
((𝐹X𝑥𝐴 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)

Theoremfidmfisupp 39206 A function with a finite domain is finitely supported. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐹:𝐷𝑅)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑍𝑉)       (𝜑𝐹 finSupp 𝑍)

Theoremmapsnend 39207 Set exponentiation to a singleton exponent is equinumerous to its base. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝑚 {𝐵}) ≈ 𝐴)

Theoremchoicefi 39208* For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)       (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))

Theoremmpct 39209 The exponentiation of a countable set to a finite set is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 ≼ ω)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → (𝐴𝑚 𝐵) ≼ ω)

Theoremcnmetcoval 39210 Value of the distance function of the metric space of complex numbers, composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐷 = (abs ∘ − )    &   (𝜑𝐹:𝐴⟶(ℂ × ℂ))    &   (𝜑𝐵𝐴)       (𝜑 → ((𝐷𝐹)‘𝐵) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))

Theoremfcomptss 39211* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐵𝐶)    &   (𝜑𝐺:𝐶𝐷)       (𝜑 → (𝐺𝐹) = (𝑥𝐴 ↦ (𝐺‘(𝐹𝑥))))

Theoremelmapsnd 39212 Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹 Fn {𝐴})    &   (𝜑𝐵𝑉)    &   (𝜑 → (𝐹𝐴) ∈ 𝐵)       (𝜑𝐹 ∈ (𝐵𝑚 {𝐴}))

Theoremmapss2 39213 Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))

Theoremfsneq 39214 Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   𝐵 = {𝐴}    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑𝐺 Fn 𝐵)       (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))

Theoremdifmap 39215 Difference of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → ((𝐴𝐵) ↑𝑚 𝐶) ⊆ ((𝐴𝑚 𝐶) ∖ (𝐵𝑚 𝐶)))

Theoremunirnmap 39216 Given a subset of a set exponentiation, the base set can be restricted. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝑋 ⊆ (𝐵𝑚 𝐴))       (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐴))

Theoreminmap 39217 Intersection of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 → ((𝐴𝑚 𝐶) ∩ (𝐵𝑚 𝐶)) = ((𝐴𝐵) ↑𝑚 𝐶))

Theoremfcoss 39218 Composition of two mappings. Similar to fco 6045, but with a weaker condition on the domain of 𝐹. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐺:𝐷𝐶)       (𝜑 → (𝐹𝐺):𝐷𝐵)

Theoremfsneqrn 39219 Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   𝐵 = {𝐴}    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑𝐺 Fn 𝐵)       (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) ∈ ran 𝐺))

Theoremdifmapsn 39220 Difference of two sets exponentiatiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 → ((𝐴𝑚 {𝐶}) ∖ (𝐵𝑚 {𝐶})) = ((𝐴𝐵) ↑𝑚 {𝐶}))

Theoremmapssbi 39221 Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))

Theoremunirnmapsn 39222 Equality theorem for a subset of a set exponentiation, where the exponent is a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝐶 = {𝐴}    &   (𝜑𝑋 ⊆ (𝐵𝑚 𝐶))       (𝜑𝑋 = (ran 𝑋𝑚 𝐶))

Theoremiunmapss 39223* The indexed union of set exponentiations is a subset of the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)       (𝜑 𝑥𝐴 (𝐵𝑚 𝐶) ⊆ ( 𝑥𝐴 𝐵𝑚 𝐶))

Theoremssmapsn 39224* A subset 𝐶 of a set exponentiation to a singleton, is its projection 𝐷 exponentiated to the singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑓𝐷    &   (𝜑𝐴𝑉)    &   (𝜑𝐶 ⊆ (𝐵𝑚 {𝐴}))    &   𝐷 = 𝑓𝐶 ran 𝑓       (𝜑𝐶 = (𝐷𝑚 {𝐴}))

Theoremiunmapsn 39225* The indexed union of set exponentiations to a singleton is equal to the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 𝑥𝐴 (𝐵𝑚 {𝐶}) = ( 𝑥𝐴 𝐵𝑚 {𝐶}))

Theoremabsfico 39226 Mapping domain and codomain of the absolute value function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
abs:ℂ⟶(0[,)+∞)

Theoremicof 39227 The set of left-closed right-open intervals of extended reals maps to subsets of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
[,):(ℝ* × ℝ*)⟶𝒫 ℝ*

Theoremrnmpt0 39228* The range of a function in map-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))

Theoremrnmptn0 39229* The range of a function in map-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ran 𝐹 ≠ ∅)

Theoremelpmrn 39230 The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → ran 𝐹𝐴)

Theoremimaexi 39231 The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝑉       (𝐴𝐵) ∈ V

Theoremaxccdom 39232* Relax the constraint on ax-cc to dominance instead of equinumerosity. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋 ≼ ω)    &   ((𝜑𝑧𝑋) → 𝑧 ≠ ∅)       (𝜑 → ∃𝑓(𝑓 Fn 𝑋 ∧ ∀𝑧𝑋 (𝑓𝑧) ∈ 𝑧))

Theoremdmmptdf 39233* The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)

Theoremelpmi2 39234 The domain of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → dom 𝐹𝐵)

Theoremdmrelrnrel 39235* A relation preserving function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑𝐵𝑅𝐶)       (𝜑 → (𝐹𝐵)𝑆(𝐹𝐶))

Theoremfdmd 39236 The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → dom 𝐹 = 𝐴)

Theoremfco3 39237 Functionality of a composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)       (𝜑 → (𝐹𝐺):(𝐺 “ dom 𝐹)⟶ran 𝐹)

Theoremdmexd 39238 The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → dom 𝐴 ∈ V)

Theoremfvcod 39239 Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑 → Fun 𝐺)    &   (𝜑𝐴 ∈ dom 𝐺)    &   𝐻 = (𝐹𝐺)       (𝜑 → (𝐻𝐴) = (𝐹‘(𝐺𝐴)))

Theoremfcod 39240 Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐵𝐶)    &   (𝜑𝐺:𝐴𝐵)       (𝜑 → (𝐹𝐺):𝐴𝐶)

Theoremfreld 39241 A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → Rel 𝐹)

Theoremfrnd 39242 The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → ran 𝐹𝐵)

Theoremelrnmpt2id 39243* Membership in the range of an operation class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝐶𝑉) → (𝑥𝐹𝑦) ∈ ran 𝐹)

Theoremfvmptelrn 39244* A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑 → (𝑥𝐴𝐵):𝐴𝐶)       ((𝜑𝑥𝐴) → 𝐵𝐶)

Theoremaxccd 39245* An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ≈ ω)    &   ((𝜑𝑥𝐴) → 𝑥 ≠ ∅)       (𝜑 → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝑥)

Theoremaxccd2 39246* An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ≼ ω)    &   ((𝜑𝑥𝐴) → 𝑥 ≠ ∅)       (𝜑 → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝑥)

Theoremfunimassd 39247* Sufficient condition for the image of a function being a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → Fun 𝐹)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)       (𝜑 → (𝐹𝐴) ⊆ 𝐵)

Theoremfimassd 39248 The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝑋) ⊆ 𝐵)

Theoremfeqresmptf 39249* Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))

Theoremfnmptd 39250* The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹 Fn 𝐴)

Theoremelrnmpt1d 39251 Elementhood in an image set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝑥𝐴)    &   (𝜑𝐵𝑉)       (𝜑𝐵 ∈ ran 𝐹)

Theoremdmresss 39252 The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
dom (𝐴𝐵) ⊆ dom 𝐴

Theoremmptima 39253* Image of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)

Theoremdmmptssf 39254 The domain of a mapping is a subset of its base class. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐹 = (𝑥𝐴𝐵)       dom 𝐹𝐴

Theoremdmmptdf2 39255 The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑥𝐵    &   𝐴 = (𝑥𝐵𝐶)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)       (𝜑 → dom 𝐴 = 𝐵)

Theoremdmuz 39256 Domain of the upper integers function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
dom ℤ = ℤ

Theoremfndmd 39257 The domain of a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)       (𝜑 → dom 𝐹 = 𝐴)

Theoremfmptd2f 39258* Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)

Theoremmpteq1df 39259 An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))

Theoremmptexf 39260 If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6469. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V

Theoremfvmptd2 39261* Deduction version of fvmpt 6269. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐷𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremfvmpt4 39262* Value of a function given by the "maps to" notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)

Theoremfvmptd3 39263* Deduction version of fvmpt 6269. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐷𝐵)    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremfmptf 39264* Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐵    &   𝐹 = (𝑥𝐴𝐶)       (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)

Theoremresimass 39265 The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)

Theoremmptssid 39266 The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐶 = {𝑥𝐴𝐵 ∈ V}       (𝑥𝐴𝐵) = (𝑥𝐶𝐵)

Theoremmptfnd 39267 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)

Theoremmpteq12da 39268 An equality inference for the maps to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremrnmptlb 39269* Boundness below of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧)

Theoremelpreimad 39270 Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑 → (𝐹𝐵) ∈ 𝐶)       (𝜑𝐵 ∈ (𝐹𝐶))

Theoremrnmptbddlem 39271* Boundness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)

Theoremrnmptbdd 39272* Boundness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)

Theoremmptima2 39273* Image of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐶𝐴)       (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))

Theoremfvelimad 39274* Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐹    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐶 ∈ (𝐹𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶)

Theoremfnfvimad 39275 A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))

Theoremfmptd2 39276* Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)

Theoremfunimaeq 39277* Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝐴 ⊆ dom 𝐹)    &   (𝜑𝐴 ⊆ dom 𝐺)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theoremrnmptssf 39278* The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)

Theoremrnmptbd2lem 39279* Boundness below of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))

Theoremrnmptbd2 39280* Boundness below of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑦𝑧))

Theoreminfnsuprnmpt 39281* The indexed infimum of real numbers is the negative of the indexed supremum of the negative values. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝑦𝐵)       (𝜑 → inf(ran (𝑥𝐴𝐵), ℝ, < ) = -sup(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))

Theoremsuprclrnmpt 39282* Closure of the indexed supremum of a nonempty bounded set of reals. Range of a function in map-to notation can be used, to express an indexed supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 ≠ ∅)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) ∈ ℝ)

Theoremsuprubrnmpt2 39283* A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷 ∈ ℝ)    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝜑𝐷 ≤ sup(ran (𝑥𝐴𝐵), ℝ, < ))

Theoremsuprubrnmpt 39284* A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)       ((𝜑𝑥𝐴) → 𝐵 ≤ sup(ran (𝑥𝐴𝐵), ℝ, < ))

Theoremrnmptssdf 39285* The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑥𝐶    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → ran 𝐹𝐶)

Theoremrnmptbdlem 39286* Boundness above of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   𝑦𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))

Theoremrnmptbd 39287* Boundness above of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))

Theoremrnmptss2 39288* The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)       (𝜑 → ran (𝑥𝐴𝐶) ⊆ ran (𝑥𝐵𝐶))

Theoremelmptima 39289* The image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝐶𝑉 → (𝐶 ∈ ((𝑥𝐴𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴𝐷)𝐶 = 𝐵))

Theoremralrnmpt3 39290* A restricted quantifier over an image set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (𝜑 → (∀𝑦 ∈ ran (𝑥𝐴𝐵)𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremfvelima2 39291* Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)

Theoremfunresd 39292 A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑 → Fun 𝐹)       (𝜑 → Fun (𝐹𝐴))

Theoremrnmptssbi 39293* The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (ran 𝐹𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))

Theoremfnfvima2 39294 Given an element of the preimage, its function value is in the image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))

Theoremfnfvelrnd 39295 A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐹𝐵) ∈ ran 𝐹)

Theoremimass2d 39296 Subset theorem for image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremimassmpt 39297* Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝐵𝑉)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑 → ((𝐹𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴𝐶)𝐵𝐷))

20.31.3  Ordering on real numbers - Real and complex numbers basic operations

Theoremsub2times 39298 Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℂ → (𝐴 − (2 · 𝐴)) = -𝐴)

Theoremxrltled 39299 'Less than' implies 'less than or equal to', for extended reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴𝐵)

Theoremabssubrp 39300 The distance of two distinct complex number is a strictly positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴𝐵) → (abs‘(𝐴𝐵)) ∈ ℝ+)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >