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Theorem List for Metamath Proof Explorer - 43701-43800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfatprc 43701 A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)

Theoremdfatelrn 43702 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)

20.41.4.5  Alternative definition of the value of a function

Theoremdfafv2 43703 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 43704 Equality deduction for function value, analogous to fveq12d 6652. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))

Theoremafveq1 43705 Equality theorem for function value, analogous to fveq1 6644. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))

Theoremafveq2 43706 Equality theorem for function value, analogous to fveq1 6644. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))

Theoremnfafv 43707 Bound-variable hypothesis builder for function value, analogous to nffv 6655. To prove a deduction version of this analogous to nffvd 6657 is not easily possible because a deduction version of nfdfat 43698 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹'''𝐴)

Theoremcsbafv12g 43708 Move class substitution in and out of a function value, analogous to csbfv12 6688, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7177. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))

Theoremafvfundmfveq 43709 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 43710 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 43711 The value of a class outside its domain is the universe, compare with ndmfv 6675. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Theoremafvvdm 43712 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 43713 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6682. (Contributed by Alexander van der Vekens, 25-May-2017.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Theoremafvvfunressn 43714 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 43715 A function's value at a proper class is the universe, compare with fvprc 6638. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ V → (𝐹'''𝐴) = V)

Theoremafvvv 43716 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 43717 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 43718 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 43719 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 43720 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 43721 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 43722 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 43723 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 43724* The value of a function at a unique point, analogous to fveu 6636. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})

Theoremfnbrafvb 43725 Equivalence of function value and binary relation, analogous to fnbrfvb 6693. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))

Theoremfnopafvb 43726 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6694. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Theoremfunbrafvb 43727 Equivalence of function value and binary relation, analogous to funbrfvb 6695. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunopafvb 43728 Equivalence of function value and ordered pair membership, analogous to funopfvb 6696. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

Theoremfunbrafv 43729 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6691. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Theoremfunbrafv2b 43730 Function value in terms of a binary relation, analogous to funbrfv2b 6698. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))

Theoremdfafn5a 43731* Representation of a function in terms of its values, analogous to dffn5 6699 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))

Theoremdfafn5b 43732* Representation of a function in terms of its values, analogous to dffn5 6699 (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 43733* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6700. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})

Theoremafvelrnb 43734* A member of a function's range is a value of the function, analogous to fvelrnb 6701 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 43735* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6701. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremdfaimafn 43736* Alternate definition of the image of a function, analogous to dfimafn 6703. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})

Theoremdfaimafn2 43737* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6704. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})

Theoremafvelima 43738* Function value in an image, analogous to fvelima 6706. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹'''𝑥) = 𝐴)

Theoremafvelrn 43739 A function's value belongs to its range, analogous to fvelrn 6821. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)

Theoremfnafvelrn 43740 A function's value belongs to its range, analogous to fnfvelrn 6825. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Theoremfafvelrn 43741 A function's value belongs to its codomain, analogous to ffvelrn 6826. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Theoremffnafv 43742* A function maps to a class to which all values belong, analogous to ffnfv 6859. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))

Theoremafvres 43743 The value of a restricted function, analogous to fvres 6664. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))

Theoremtz6.12-afv 43744* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6668. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)

Theoremtz6.12-1-afv 43745* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6667. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)

Theoremdmfcoafv 43746 Domains of a function composition, analogous to dmfco 6734. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))

Theoremafvco2 43747 Value of a function composition, analogous to fvco2 6735. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Theoremrlimdmafv 43748 Two ways to express that a function has a limit, analogous to rlimdm 14902. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (𝐹 ∈ dom ⇝𝑟𝐹𝑟 ( ⇝𝑟 '''𝐹)))

20.41.4.6  Alternative definition of the value of an operation

Theoremaoveq123d 43749 Equality deduction for operation value, analogous to oveq123d 7156. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Theoremnfaov 43750 Bound-variable hypothesis builder for operation value, analogous to nfov 7165. To prove a deduction version of this analogous to nfovd 7164 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 43707). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥 ((𝐴𝐹𝐵))

Theoremcsbaovg 43751 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 43752 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 43753 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 43754 The value of an operation outside its domain, analogous to ndmafv 43711. (Contributed by Alexander van der Vekens, 26-May-2017.)
(¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)

Theoremndmaovg 43755 The value of an operation outside its domain, analogous to ndmovg 7312. (Contributed by Alexander van der Vekens, 26-May-2017.)
((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ((𝐴𝐹𝐵)) = V)

Theoremaovvdm 43756 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 43757 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 43758 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 43759 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7173. (Contributed by Alexander van der Vekens, 26-May-2017.)
Rel dom 𝐹       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)

Theoremaovrcl 43760 Reverse closure for an operation value, analogous to afvvv 43716. In contrast to ovrcl 7176, 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 43761 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 43762 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 43763 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 43764 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 43765 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 43766 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 43767 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 43768* A frequently used special case of rspc2ev 3583 for operation values, analogous to rspceov 7182. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Theoremfnotaovb 43769 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6694. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Theoremffnaov 43770* An operation maps to a class to which all values belong, analogous to ffnov 7258. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

Theoremfaovcl 43771 Closure law for an operation, analogous to fovcl 7259. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)

Theoremaovmpt4g 43772* Value of a function given by the maps-to notation, analogous to ovmpt4g 7277. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)

Theoremaoprssdm 43773* Domain of closure of an operation. In contrast to oprssdm 7310, 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 43774 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7314 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 43775 Reverse closure law, in contrast to ndmovrcl 7315 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 43776 Any operation is commutative outside its domain, analogous to ndmovcom 7316. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )

Theoremndmaovass 43777 Any operation is associative outside its domain. In contrast to ndmovass 7317 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 43778 Any operation is distributive outside its domain. In contrast to ndmovdistr 7318 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)) )) )

20.41.5  Alternative definitions of function values (2)

In the following, a second approach is followed to define function values alternately to df-afv 43691.

The current definition of the value (𝐹𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6332) assures that this value is always a set, see fex 6966. 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 6675 and fvprc 6638). "(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 43690. 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 6676).

To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 43780) 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 43782). 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 43795, 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 43783). 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 43831), but (𝐹𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 43836.

The alternate definition, however, corresponds to the current definition ((𝐹𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 43832).

With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 43798.

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 6332 of (𝐹𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6644-> afv2eq1 43787, fveq2 6645-> afv2eq2 43788, nffv 6655-> nfafv2 43789, csbfv12 6688-> csbafv212g , rlimdm 14902-> rlimdmafv2 43829, tz6.12-1 6667-> tz6.12-1-afv2 43812, fveu 6636-> afv2eu 43809.

Six theorems proved by directly using df-fv 6332 are within a mathbox (fvsb 41171, uncov 35054) or not used (rlimdmafv 43748, avril1 28255) or experimental (dfafv2 43703, dfafv22 43830).

However, the remaining 11 theorems proved by directly using df-fv 6332 are used more or less often:

* fvex 6658: 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 43784 resp. afv2ex 43785). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6658.

* fvres 6664: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 43810). 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 43810 can be used instead of fvres 6664.

* tz6.12-2 6635 (-> tz6.12-2-afv2 43808): root theorem of many theorems which have not a strict analogue, and which are used many times:

** fvprc 6638 (-> afv2prc 43797), used in 193 proofs,

** tz6.12i 6671 (-> tz6.12i-afv2 43814), used - indirectly via fvbr0 6672 and fvrn0 6673 - in 19 proofs, and in fvclss 6979 used in fvclex 7644 used in fvresex 7645 (which is not used!) and in dcomex 9860 (used in 4 proofs),

** ndmfv 6675 (-> ndmafv2nrn ), used in 124 proofs

** nfunsn 6682 (-> nfunsnafv2 ), used by fvfundmfvn0 6683 (used in 3 proofs), and dffv2 6733 (not used)

** funpartfv 33531, setrec2lem1 45237 (mathboxes)

* fv2 6640: only used by elfv 6643, which is only used by fv3 6663, which is not used.

* dffv3 6641 (-> dfafv23 ): used by dffv4 6642 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 41620), by shftval 14427 (itself used in 11 proofs), by dffv5 33510 (mathbox) and by fvco2 6735 (-> afv2co2 43828).

* fvopab5 6777: used only by ajval 28651 (not used) and by adjval 29680, which is used in adjval2 29681 (not used) and in adjbdln 29873 (used in 7 proofs).

* zsum 15069: used (via isum 15070, sum0 15072, sumss 15075 and fsumsers 15079) in 76 proofs.

* isumshft 15188: used in pserdv2 25032 (used in logtayl 25258, binomcxplemdvsum 41074) , eftlub 15456 (used in 4 proofs), binomcxplemnotnn0 41075 (used in binomcxp 41076 only) and logtayl 25258 (used in 4 proofs).

* ovtpos 7892: used in 16 proofs.

* zprod 15285: used in 3 proofs: iprod 15286, zprodn0 15287 and prodss 15295

* iprodclim3 15348: 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 6640, dffv3 6641, fvopab5 6777, zsum 15069, isumshft 15188, ovtpos 7892 and zprod 15285 are not critical or are, hopefully, also valid for the alternative definition, fvex 6658, fvres 6664 and tz6.12-2 6635 (and the theorems based on them) are essential for the current definition of function values.

Syntaxcafv2 43779 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 43688.
class (𝐹''''𝐴)

Definitiondf-afv2 43780* 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 6332, and especially ndmfv 6675), (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 43781* 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 43782 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 43783 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 43784 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 43785 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 43786 Equality deduction for function value, analogous to fveq12d 6652. (Contributed by AV, 4-Sep-2022.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Theoremafv2eq1 43787 Equality theorem for function value, analogous to fveq1 6644. (Contributed by AV, 4-Sep-2022.)
(𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))

Theoremafv2eq2 43788 Equality theorem for function value, analogous to fveq2 6645. (Contributed by AV, 4-Sep-2022.)
(𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))

Theoremnfafv2 43789 Bound-variable hypothesis builder for function value, analogous to nffv 6655. To prove a deduction version of this analogous to nffvd 6657 is not easily possible because a deduction version of nfdfat 43698 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹''''𝐴)

Theoremcsbafv212g 43790 Move class substitution in and out of a function value, analogous to csbfv12 6688, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7177. (Contributed by AV, 4-Sep-2022.)
(𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))

Theoremfexafv2ex 43791 The alternate function value is always a set if the function (resp. the domain of the function) is a set. (Contributed by AV, 3-Sep-2022.)
(𝐹𝑉 → (𝐹''''𝐴) ∈ V)

Theoremndfatafv2nrn 43792 The alternate function value at a class 𝐴 at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022.)
𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremndmafv2nrn 43793 The value of a class outside its domain is not in the range, compare with ndmfv 6675. (Contributed by AV, 2-Sep-2022.)
𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremfunressndmafv2rn 43794 The alternate function value at a class 𝐴 is defined, i.e., in the range of the function if the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
(𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)

Theoremafv2ndefb 43795 Two ways to say that an alternate function value is not defined. (Contributed by AV, 5-Sep-2022.)
((𝐹''''𝐴) = 𝒫 ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹)

Theoremnfunsnafv2 43796 If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6682. (Contributed by AV, 2-Sep-2022.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremafv2prc 43797 A function's value at a proper class is not defined, compare with fvprc 6638. (Contributed by AV, 5-Sep-2022.)
𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremdfatafv2rnb 43798 The alternate function value at a class 𝐴 is defined, i.e. in the range of the function, iff the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
(𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)

Theoremafv2orxorb 43799 If a set is in the range of a function, the alternate function value at a class 𝐴 equals this set or is not in the range of the function iff the alternate function value at the class 𝐴 either equals this set or is not in the range of the function. If 𝐵 ∉ ran 𝐹, both disjuncts of the exclusive or can be true: (𝐹''''𝐴) = 𝐵 → (𝐹''''𝐴) ∉ ran 𝐹. (Contributed by AV, 11-Sep-2022.)
(𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))

Theoremdmafv2rnb 43800 The alternate function value at a class 𝐴 is defined, i.e., in the range of the function, iff 𝐴 is in the domain of the function. (Contributed by AV, 3-Sep-2022.)
(Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))

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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 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45347
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