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Theorem List for Metamath Proof Explorer - 39701-39800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvmap 39701 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 39702* 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 39703* Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
((𝐹X𝑥𝐴 𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
 
Theoremfidmfisupp 39704 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 39705 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 39706* 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 39707 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 39708 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 39709* 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 39710 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 39711 Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))
 
Theoremfsneq 39712 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 39713 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 39714 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 39715 Intersection of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)       (𝜑 → ((𝐴𝑚 𝐶) ∩ (𝐵𝑚 𝐶)) = ((𝐴𝐵) ↑𝑚 𝐶))
 
Theoremfcoss 39716 Composition of two mappings. Similar to fco 6096, 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 39717 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 39718 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 39719 Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑍)    &   (𝜑𝐶 ≠ ∅)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))
 
Theoremunirnmapsn 39720 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 39721* 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 39722* 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 39723* 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 39724 Mapping domain and codomain of the absolute value function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
abs:ℂ⟶(0[,)+∞)
 
Theoremicof 39725 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 39726* 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 39727* 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 39728 The range of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → ran 𝐹𝐴)
 
Theoremimaexi 39729 The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴𝑉       (𝐴𝐵) ∈ V
 
Theoremaxccdom 39730* 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 39731* 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 39732 The domain of a partial function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ (𝐴pm 𝐵) → dom 𝐹𝐵)
 
Theoremdmrelrnrel 39733* 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 39734 The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → dom 𝐹 = 𝐴)
 
Theoremfco3 39735 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 39736 The domain of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)       (𝜑 → dom 𝐴 ∈ V)
 
Theoremfvcod 39737 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 39738 Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐵𝐶)    &   (𝜑𝐺:𝐴𝐵)       (𝜑 → (𝐹𝐺):𝐴𝐶)
 
Theoremfreld 39739 A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → Rel 𝐹)
 
Theoremfrnd 39740 The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → ran 𝐹𝐵)
 
Theoremelrnmpt2id 39741* 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 39742* A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑 → (𝑥𝐴𝐵):𝐴𝐶)       ((𝜑𝑥𝐴) → 𝐵𝐶)
 
Theoremaxccd 39743* 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 39744* 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 39745* 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 39746 The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝑋) ⊆ 𝐵)
 
Theoremfeqresmptf 39747* Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
 
Theoremfnmptd 39748* 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 39749 Elementhood in an image set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝑥𝐴)    &   (𝜑𝐵𝑉)       (𝜑𝐵 ∈ ran 𝐹)
 
Theoremdmresss 39750 The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
dom (𝐴𝐵) ⊆ dom 𝐴
 
Theoremmptima 39751* Image of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
 
Theoremdmmptssf 39752 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 39753 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 39754 Domain of the upper integers function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
dom ℤ = ℤ
 
Theoremfndmd 39755 The domain of a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)       (𝜑 → dom 𝐹 = 𝐴)
 
Theoremfmptd2f 39756* Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
 
Theoremmpteq1df 39757 An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 
Theoremmptexf 39758 If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6525. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V
 
Theoremfvmptd2 39759* Deduction version of fvmpt 6321. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐷𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)
 
Theoremfvmpt4 39760* Value of a function given by the "maps to" notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
 
Theoremfvmptd3 39761* Deduction version of fvmpt 6321. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐹 = (𝑥𝐷𝐵)    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)
 
Theoremfmptf 39762* Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐵    &   𝐹 = (𝑥𝐴𝐶)       (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
 
Theoremresimass 39763 The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)
 
Theoremmptssid 39764 The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐴    &   𝐶 = {𝑥𝐴𝐵 ∈ V}       (𝑥𝐴𝐵) = (𝑥𝐶𝐵)
 
Theoremmptfnd 39765 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 39766 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 39767* 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 39768 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 39769* 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 39770* 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 39771* Image of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐶𝐴)       (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
 
Theoremfvelimad 39772* Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝑥𝐹    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐶 ∈ (𝐹𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶)
 
Theoremfnfvimad 39773 A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))
 
Theoremfmptd2 39774* Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
 
Theoremfunimaeq 39775* 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 39776* 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 39777* 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 39778* 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 39779* 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 39780* 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 39781* 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 39782* 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 39783* 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 39784* 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 39785* 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 39786* 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 39787* 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 39788* 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 39789* Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
 
Theoremfunresd 39790 A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑 → Fun 𝐹)       (𝜑 → Fun (𝐹𝐴))
 
Theoremrnmptssbi 39791* 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 39792 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 39793 A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐹𝐵) ∈ ran 𝐹)
 
Theoremimass2d 39794 Subset theorem for image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremimassmpt 39795* 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))
 
Theoremfnssresd 39796 Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐹𝐵) Fn 𝐵)
 
Theoremfpmd 39797 A total function is a partial function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝐴)    &   (𝜑𝐹:𝐶𝐵)       (𝜑𝐹 ∈ (𝐵pm 𝐴))
 
Theoremfconst7 39798* An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑥𝜑    &   𝑥𝐹    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝑉)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)       (𝜑𝐹 = (𝐴 × {𝐵}))
 
20.32.3  Ordering on real numbers - Real and complex numbers basic operations
 
Theoremsub2times 39799 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 39800 '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)
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