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Theorem List for Metamath Proof Explorer - 43201-43300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcafv 43201 Extend the definition of a class to include the value of a function. Read: "the value of 𝐹 at 𝐴 " or "𝐹 of 𝐴". In a previous version, the symbol " ' " was used. However, since the similarity with the symbol used for the current definition of a function's value (see df-fv 6362), which, by the way, was intended to visualize that in many cases and " ' " are exchangeable, makes reading the theorems, especially those which uses both definitions as dfafv2 43216, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 6362 and df-ima 5567. And not three backticks ( three times ) since that would be annoying to escape in a comment. (See remark of Norman Megill and Gerard Lang at https://groups.google.com/g/metamath/c/cteNUppB6A4 5567).
class (𝐹'''𝐴)
 
Syntaxcaov 43202 Extend class notation to include the value of an operation 𝐹 (such as +) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by a pair of parentheses in contrast to the current definition, see df-ov 7153.
class ((𝐴𝐹𝐵))
 
Definitiondf-dfat 43203 Definition of the predicate that determines if some class 𝐹 is defined as function for an argument 𝐴 or, in other words, if the function value for some class 𝐹 for an argument 𝐴 is defined. We say that 𝐹 is defined at 𝐴 if a 𝐹 is a function restricted to the member 𝐴 of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
 
Definitiondf-afv 43204* Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹𝐴) = ∅ (see df-fv 6362 and ndmfv 6699), (𝐹'''𝐴) = V if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.) (Revised by BJ/AV, 25-Aug-2022.)
(𝐹'''𝐴) = (℩'𝑥𝐴𝐹𝑥)
 
Definitiondf-aov 43205 Define the value of an operation. In contrast to df-ov 7153, the alternative definition for a function value (see df-afv 43204) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
 
20.40.4.1  Restricted quantification (extension)
 
Theoremralbinrald 43206* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
(𝜑𝑋𝐴)    &   (𝑥𝐴𝑥 = 𝑋)    &   (𝑥 = 𝑋 → (𝜓𝜃))       (𝜑 → (∀𝑥𝐴 𝜓𝜃))
 
20.40.4.2  The universal class (extension)
 
Theoremnvelim 43207 If a class is the universal class it doesn't belong to any class, generalization of nvel 5217. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐴 = V → ¬ 𝐴𝐵)
 
20.40.4.3  Introduce the Axiom of Power Sets (extension)
 
Theoremalneu 43208 If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.)
(∀𝑥𝜑 → ¬ ∃!𝑥𝜑)
 
Theoremeu2ndop1stv 43209* If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)
 
20.40.4.4  Predicate "defined at"
 
Theoremdfateq12d 43210 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
 
Theoremnfdfat 43211 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g., for Fun/Rel, dom, , etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 defAt 𝐴
 
Theoremdfdfat2 43212* Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
 
Theoremfundmdfat 43213 A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
 
Theoremdfatprc 43214 A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)
 
Theoremdfatelrn 43215 The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐹 is defined at 𝐴. (Contributed by AV, 1-Sep-2022.)
(𝐹 defAt 𝐴 → (𝐹𝐴) ∈ ran 𝐹)
 
20.40.4.5  Alternative definition of the value of a function
 
Theoremdfafv2 43216 Alternative definition of (𝐹'''𝐴) using (𝐹𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.) (Revised by AV, 25-Aug-2022.)
(𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
 
Theoremafveq12d 43217 Equality deduction for function value, analogous to fveq12d 6676. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))
 
Theoremafveq1 43218 Equality theorem for function value, analogous to fveq1 6668. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))
 
Theoremafveq2 43219 Equality theorem for function value, analogous to fveq1 6668. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))
 
Theoremnfafv 43220 Bound-variable hypothesis builder for function value, analogous to nffv 6679. To prove a deduction version of this analogous to nffvd 6681 is not easily possible because a deduction version of nfdfat 43211 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹'''𝐴)
 
Theoremcsbafv12g 43221 Move class substitution in and out of a function value, analogous to csbfv12 6712, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7192. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
 
Theoremafvfundmfveq 43222 If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
 
Theoremafvnfundmuv 43223 If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
 
Theoremndmafv 43224 The value of a class outside its domain is the universe, compare with ndmfv 6699. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)
 
Theoremafvvdm 43225 If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ dom 𝐹)
 
Theoremnfunsnafv 43226 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6706. (Contributed by Alexander van der Vekens, 25-May-2017.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)
 
Theoremafvvfunressn 43227 If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴}))
 
Theoremafvprc 43228 A function's value at a proper class is the universe, compare with fvprc 6662. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ V → (𝐹'''𝐴) = V)
 
Theoremafvvv 43229 If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ V)
 
Theoremafvpcfv0 43230 If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
 
Theoremafvnufveq 43231 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
 
Theoremafvvfveq 43232 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵 → (𝐹'''𝐴) = (𝐹𝐴))
 
Theoremafv0fv0 43233 If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)
 
Theoremafvfvn0fveq 43234 If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹𝐴))
 
Theoremafv0nbfvbi 43235 The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
(∅ ∉ 𝐵 → ((𝐹'''𝐴) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))
 
Theoremafvfv0bi 43236 The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
 
Theoremafveu 43237* The value of a function at a unique point, analogous to fveu 6660. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
 
Theoremfnbrafvb 43238 Equivalence of function value and binary relation, analogous to fnbrfvb 6717. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))
 
Theoremfnopafvb 43239 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6718. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
 
Theoremfunbrafvb 43240 Equivalence of function value and binary relation, analogous to funbrfvb 6719. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))
 
Theoremfunopafvb 43241 Equivalence of function value and ordered pair membership, analogous to funopfvb 6720. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
 
Theoremfunbrafv 43242 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6715. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))
 
Theoremfunbrafv2b 43243 Function value in terms of a binary relation, analogous to funbrfv2b 6722. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))
 
Theoremdfafn5a 43244* Representation of a function in terms of its values, analogous to dffn5 6723 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
 
Theoremdfafn5b 43245* Representation of a function in terms of its values, analogous to dffn5 6723 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
(∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥))))
 
Theoremfnrnafv 43246* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6724. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
 
Theoremafvelrnb 43247* A member of a function's range is a value of the function, analogous to fvelrnb 6725 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
 
Theoremafvelrnb0 43248* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6725. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
 
Theoremdfaimafn 43249* Alternate definition of the image of a function, analogous to dfimafn 6727. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})
 
Theoremdfaimafn2 43250* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6728. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})
 
Theoremafvelima 43251* Function value in an image, analogous to fvelima 6730. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹'''𝑥) = 𝐴)
 
Theoremafvelrn 43252 A function's value belongs to its range, analogous to fvelrn 6842. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)
 
Theoremfnafvelrn 43253 A function's value belongs to its range, analogous to fnfvelrn 6846. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)
 
Theoremfafvelrn 43254 A function's value belongs to its codomain, analogous to ffvelrn 6847. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)
 
Theoremffnafv 43255* A function maps to a class to which all values belong, analogous to ffnfv 6880. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
 
Theoremafvres 43256 The value of a restricted function, analogous to fvres 6688. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
 
Theoremtz6.12-afv 43257* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6692. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)
 
Theoremtz6.12-1-afv 43258* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6691. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)
 
Theoremdmfcoafv 43259 Domains of a function composition, analogous to dmfco 6756. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))
 
Theoremafvco2 43260 Value of a function composition, analogous to fvco2 6757. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
 
Theoremrlimdmafv 43261 Two ways to express that a function has a limit, analogous to rlimdm 14903. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (𝐹 ∈ dom ⇝𝑟𝐹𝑟 ( ⇝𝑟 '''𝐹)))
 
20.40.4.6  Alternative definition of the value of an operation
 
Theoremaoveq123d 43262 Equality deduction for operation value, analogous to oveq123d 7171. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )
 
Theoremnfaov 43263 Bound-variable hypothesis builder for operation value, analogous to nfov 7180. To prove a deduction version of this analogous to nfovd 7179 is not quickly possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of alternative operation values is based on are not available (see nfafv 43220). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥 ((𝐴𝐹𝐵))
 
Theoremcsbaovg 43264 Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐴𝐷𝐴 / 𝑥 ((𝐵𝐹𝐶)) = ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐹𝐴 / 𝑥𝐶)) )
 
Theoremaovfundmoveq 43265 If a class is a function restricted to an ordered pair of its domain, then the value of the operation on this pair is equal for both definitions. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
 
Theoremaovnfundmuv 43266 If an ordered pair is not in the domain of a class or the class is not a function restricted to the ordered pair, then the operation value for this pair is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = V)
 
Theoremndmaov 43267 The value of an operation outside its domain, analogous to ndmafv 43224. (Contributed by Alexander van der Vekens, 26-May-2017.)
(¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
 
Theoremndmaovg 43268 The value of an operation outside its domain, analogous to ndmovg 7325. (Contributed by Alexander van der Vekens, 26-May-2017.)
((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ((𝐴𝐹𝐵)) = V)
 
Theoremaovvdm 43269 If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
 
Theoremnfunsnaov 43270 If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
(¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → ((𝐴𝐹𝐵)) = V)
 
Theoremaovvfunressn 43271 If the operation value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) ∈ 𝐶 → Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}))
 
Theoremaovprc 43272 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7188. (Contributed by Alexander van der Vekens, 26-May-2017.)
Rel dom 𝐹       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)
 
Theoremaovrcl 43273 Reverse closure for an operation value, analogous to afvvv 43229. In contrast to ovrcl 7191, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)
Rel dom 𝐹       ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theoremaovpcov0 43274 If the alternative value of the operation on an ordered pair is the universal class, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅)
 
Theoremaovnuoveq 43275 The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
 
Theoremaovvoveq 43276 The alternative value of the operation on an ordered pair equals the operation's value on this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) ∈ 𝐶 → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
 
Theoremaov0ov0 43277 If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅)
 
Theoremaovovn0oveq 43278 If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
 
Theoremaov0nbovbi 43279 The operation's value on an ordered pair is an element of a set if and only if the alternative value of the operation on this ordered pair is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
(∅ ∉ 𝐶 → ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))
 
Theoremaovov0bi 43280 The operation's value on an ordered pair is the empty set if and only if the alternative value of the operation on this ordered pair is either the empty set or the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐴𝐹𝐵) = ∅ ↔ ( ((𝐴𝐹𝐵)) = ∅ ∨ ((𝐴𝐹𝐵)) = V))
 
Theoremrspceaov 43281* A frequently used special case of rspc2ev 3639 for operation values, analogous to rspceov 7197. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )
 
Theoremfnotaovb 43282 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6718. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
 
Theoremffnaov 43283* An operation maps to a class to which all values belong, analogous to ffnov 7272. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))
 
Theoremfaovcl 43284 Closure law for an operation, analogous to fovcl 7273. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)
 
Theoremaovmpt4g 43285* Value of a function given by the maps-to notation, analogous to ovmpt4g 7291. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)
 
Theoremaoprssdm 43286* Domain of closure of an operation. In contrast to oprssdm 7323, no additional property for S (¬ ∅ ∈ 𝑆) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆)       (𝑆 × 𝑆) ⊆ dom 𝐹
 
Theoremndmaovcl 43287 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7327 where it is required that the domain contains the empty set (∅ ∈ 𝑆). (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)    &   ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)    &    ((𝐴𝐹𝐵)) ∈ V        ((𝐴𝐹𝐵)) ∈ 𝑆
 
Theoremndmaovrcl 43288 Reverse closure law, in contrast to ndmovrcl 7328 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴𝑆𝐵𝑆))
 
Theoremndmaovcom 43289 Any operation is commutative outside its domain, analogous to ndmovcom 7329. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
 
Theoremndmaovass 43290 Any operation is associative outside its domain. In contrast to ndmovass 7330 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) )
 
Theoremndmaovdistr 43291 Any operation is distributive outside its domain. In contrast to ndmovdistr 7331 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)    &   dom 𝐺 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) )
 
20.40.5  Alternative definitions of function values (2)

In the following, a second approach is followed to define function values alternately to df-afv 43204.

The current definition of the value (𝐹𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6362) assures that this value is always a set, see fex 6986. This is because this definition can be applied to any classes 𝐹 and 𝐴, and evaluates to the empty set when it is not meaningful (as shown by ndmfv 6699 and fvprc 6662). "(𝐹𝐴) is meaningful" means "the class 𝐹 regarded as function is defined at the argument 𝐴" in this context. This is also expressed by 𝐹 defAt 𝐴, see df-dfat 43203. In the theory of partial functions, it is a common case that 𝐹 is not defined at 𝐴.

Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (𝐹𝐴) = ∅ alone it cannot be decided/derived whether (𝐹𝐴) is meaningful (𝐹 is actually a function which is defined for 𝐴 and really has the function value at 𝐴) or not. Therefore, additional assumptions are required, such as ∅ ∉ ran 𝐹, ∅ ∈ ran 𝐹, 𝐹 defAt 𝐴 or Fun 𝐹𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6700).

To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 43293) would be possible which evaluates to a set not belonging to the range of 𝐹 ((𝐹''''𝐴) = 𝒫 ran 𝐹) if it is not meaningful (see ndfatafv2 43295). We say "(𝐹''''𝐴) is not defined (or undefined)" if (𝐹''''𝐴) is not in the range of 𝐹 ((𝐹''''𝐴) ∉ ran 𝐹). Because of afv2ndefb 43308, this is equivalent to ((𝐹''''𝐴) = 𝒫 ran 𝐹. If (𝐹''''𝐴) is in the range of 𝐹 ((𝐹''''𝐴) ∈ ran 𝐹), we say that "(𝐹''''𝐴) is defined".

If ran 𝐹 is a set, we can use the symbol Undef to express that (𝐹''''𝐴) is not defined: (𝐹''''𝐴) = (Undef‘ran 𝐹) (see ndfatafv2undef 43296). We could have used this symbol directly to define the alternate value of a function, which would have the advantage that (𝐹''''𝐴) would always be a set. But first this symbol is defined using the original function value, which would not make it possible to replace the original definition by the alternate definition, and second we would have to assume that ran 𝐹 ∈ V in most of the theorems.

To summarize, that means (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹𝐴) = ∅ (see afv2ndeffv0 43344), but (𝐹𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 43349.

The alternate definition, however, corresponds to the current definition ((𝐹𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 43345).

With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 43311.

An interesting question would be if (𝐹𝐴) could be replaced by (𝐹'''𝐴) in most of the theorems based on function values. If we look at the (currently 24) proofs using the definition df-fv 6362 of (𝐹𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6668-> afv2eq1 43300, fveq2 6669-> afv2eq2 43301, nffv 6679-> nfafv2 43302, csbfv12 6712-> csbafv212g , rlimdm 14903-> rlimdmafv2 43342, tz6.12-1 6691-> tz6.12-1-afv2 43325, fveu 6660-> afv2eu 43322.

Six theorems proved by directly using df-fv 6362 are within a mathbox (fvsb 40668, uncov 34759) or not used (rlimdmafv 43261, avril1 28175) or experimental (dfafv2 43216, dfafv22 43343).

However, the remaining 11 theorems proved by directly using df-fv 6362 are used more or less often:

* fvex 6682: used in about 1600 proofs: Only if the function is defined at the argument, or the range of the function/class is a set, analog theorems can be proven (dfatafv2ex 43297 resp. afv2ex 43298). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6682.

* fvres 6688: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 43323). In the undefined case such a theorem cannot exist (without additional assumtions), because the range of (𝐹𝐵) is mostly different from the range of 𝐹, and therefore also the "undefined" values are different. All of these 400 proofs have to be checked if afv2res 43323 can be used instead of fvres 6688.

* tz6.12-2 6659 (-> tz6.12-2-afv2 43321): root theorem of many theorems which have not a strict analogue, and which are used many times:

** fvprc 6662 (-> afv2prc 43310), used in 193 proofs,

** tz6.12i 6695 (-> tz6.12i-afv2 43327), used - indirectly via fvbr0 6696 and fvrn0 6697 - in 19 proofs, and in fvclss 6997 used in fvclex 7656 used in fvresex 7657 (which is not used!) and in dcomex 9863 (used in 4 proofs),

** ndmfv 6699 (-> ndmafv2nrn ), used in 124 proofs

** nfunsn 6706 (-> nfunsnafv2 ), used by fvfundmfvn0 6707 (used in 3 proofs), and dffv2 6755 (not used)

** funpartfv 33309, setrec2lem1 44698 (mathboxes)

* fv2 6664: only used by elfv 6667, which is only used by fv3 6687, which is not used.

* dffv3 6665 (-> dfafv23 ): used by dffv4 6666 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 41117), by shftval 14428 (itself used in 11 proofs), by dffv5 33288 (mathbox) and by fvco2 6757 (-> afv2co2 43341).

* fvopab5 6798: used only by ajval 28571 (not used) and by adjval 29600, which is used in adjval2 29601 (not used) and in adjbdln 29793 (used in 7 proofs).

* zsum 15070: used (via isum 15071, sum0 15073, sumss 15076 and fsumsers 15080) in 76 proofs.

* isumshft 15189: used in pserdv2 24952 (used in logtayl 25175, binomcxplemdvsum 40571) , eftlub 15457 (used in 4 proofs), binomcxplemnotnn0 40572 (used in binomcxp 40573 only) and logtayl 25175 (used in 4 proofs).

* ovtpos 7903: used in 16 proofs.

* zprod 15286: used in 3 proofs: iprod 15287, zprodn0 15288 and prodss 15296

* iprodclim3 15349: not used!

As a result of this analysis we can say that the current definition of a function value is crucial for Metamath and cannot be exchanged easily with an alternative definition. While fv2 6664, dffv3 6665, fvopab5 6798, zsum 15070, isumshft 15189, ovtpos 7903 and zprod 15286 are not critical or are, hopefully, also valid for the alternative definition, fvex 6682, fvres 6688 and tz6.12-2 6659 (and the theorems based on them) are essential for the current definition of function values.

 
Syntaxcafv2 43292 Extend the definition of a class to include the alternate function value. Read: "the value of 𝐹 at 𝐴 " or "𝐹 of 𝐴". For using several apostrophes as a symbol see comment for cafv 43201.
class (𝐹''''𝐴)
 
Definitiondf-afv2 43293* Alternate definition of the value of a function, (𝐹''''𝐴), also known as function application (and called "alternate function value" in the following). In contrast to (𝐹𝐴) = ∅ (see comment of df-fv 6362, and especially ndmfv 6699), (𝐹''''𝐴) is guaranteed not to be in the range of 𝐹 if 𝐹 is not defined at 𝐴 (whereas can be a member of ran 𝐹). (Contributed by AV, 2-Sep-2022.)
(𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
 
Theoremdfatafv2iota 43294* If a function is defined at a class 𝐴 the alternate function value at 𝐴 is the unique value assigned to 𝐴 by the function (analogously to (𝐹𝐴)). (Contributed by AV, 2-Sep-2022.)
(𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
 
Theoremndfatafv2 43295 The alternate function value at a class 𝐴 if the function is not defined at this set 𝐴. (Contributed by AV, 2-Sep-2022.)
𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
 
Theoremndfatafv2undef 43296 The alternate function value at a class 𝐴 is undefined if the function, whose range is a set, is not defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
((ran 𝐹𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹))
 
Theoremdfatafv2ex 43297 The alternate function value at a class 𝐴 is always a set if the function/class 𝐹 is defined at 𝐴. (Contributed by AV, 6-Sep-2022.)
(𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
 
Theoremafv2ex 43298 The alternate function value is always a set if the range of the function is a set. (Contributed by AV, 2-Sep-2022.)
(ran 𝐹𝑉 → (𝐹''''𝐴) ∈ V)
 
Theoremafv2eq12d 43299 Equality deduction for function value, analogous to fveq12d 6676. (Contributed by AV, 4-Sep-2022.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
 
Theoremafv2eq1 43300 Equality theorem for function value, analogous to fveq1 6668. (Contributed by AV, 4-Sep-2022.)
(𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))
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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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44808
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