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Theorem List for Metamath Proof Explorer - 44101-44200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfdfat2 44101* 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 44102 A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Theoremdfatprc 44103 A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)

Theoremdfatelrn 44104 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 44105 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 44106 Equality deduction for function value, analogous to fveq12d 6670. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))

Theoremafveq1 44107 Equality theorem for function value, analogous to fveq1 6662. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))

Theoremafveq2 44108 Equality theorem for function value, analogous to fveq1 6662. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))

Theoremnfafv 44109 Bound-variable hypothesis builder for function value, analogous to nffv 6673. To prove a deduction version of this analogous to nffvd 6675 is not easily possible because a deduction version of nfdfat 44100 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹'''𝐴)

Theoremcsbafv12g 44110 Move class substitution in and out of a function value, analogous to csbfv12 6706, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7198. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))

Theoremafvfundmfveq 44111 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 44112 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 44113 The value of a class outside its domain is the universe, compare with ndmfv 6693. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Theoremafvvdm 44114 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 44115 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6700. (Contributed by Alexander van der Vekens, 25-May-2017.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Theoremafvvfunressn 44116 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 44117 A function's value at a proper class is the universe, compare with fvprc 6655. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ V → (𝐹'''𝐴) = V)

Theoremafvvv 44118 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 44119 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 44120 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 44121 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 44122 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 44123 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 44124 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 44125 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 44126* The value of a function at a unique point, analogous to fveu 6653. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})

Theoremfnbrafvb 44127 Equivalence of function value and binary relation, analogous to fnbrfvb 6711. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))

Theoremfnopafvb 44128 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6712. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Theoremfunbrafvb 44129 Equivalence of function value and binary relation, analogous to funbrfvb 6713. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunopafvb 44130 Equivalence of function value and ordered pair membership, analogous to funopfvb 6714. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

Theoremfunbrafv 44131 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6709. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Theoremfunbrafv2b 44132 Function value in terms of a binary relation, analogous to funbrfv2b 6716. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))

Theoremdfafn5a 44133* Representation of a function in terms of its values, analogous to dffn5 6717 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))

Theoremdfafn5b 44134* Representation of a function in terms of its values, analogous to dffn5 6717 (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 44135* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6718. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})

Theoremafvelrnb 44136* A member of a function's range is a value of the function, analogous to fvelrnb 6719 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 44137* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6719. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremdfaimafn 44138* Alternate definition of the image of a function, analogous to dfimafn 6721. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})

Theoremdfaimafn2 44139* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6722. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})

Theoremafvelima 44140* Function value in an image, analogous to fvelima 6724. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹'''𝑥) = 𝐴)

Theoremafvelrn 44141 A function's value belongs to its range, analogous to fvelrn 6841. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)

Theoremfnafvelrn 44142 A function's value belongs to its range, analogous to fnfvelrn 6845. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Theoremfafvelrn 44143 A function's value belongs to its codomain, analogous to ffvelrn 6846. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Theoremffnafv 44144* A function maps to a class to which all values belong, analogous to ffnfv 6879. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))

Theoremafvres 44145 The value of a restricted function, analogous to fvres 6682. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))

Theoremtz6.12-afv 44146* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6686. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)

Theoremtz6.12-1-afv 44147* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6685. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)

Theoremdmfcoafv 44148 Domains of a function composition, analogous to dmfco 6753. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))

Theoremafvco2 44149 Value of a function composition, analogous to fvco2 6754. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Theoremrlimdmafv 44150 Two ways to express that a function has a limit, analogous to rlimdm 14969. (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 44151 Equality deduction for operation value, analogous to oveq123d 7177. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Theoremnfaov 44152 Bound-variable hypothesis builder for operation value, analogous to nfov 7186. To prove a deduction version of this analogous to nfovd 7185 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 44109). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥 ((𝐴𝐹𝐵))

Theoremcsbaovg 44153 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 44154 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 44155 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 44156 The value of an operation outside its domain, analogous to ndmafv 44113. (Contributed by Alexander van der Vekens, 26-May-2017.)
(¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)

Theoremndmaovg 44157 The value of an operation outside its domain, analogous to ndmovg 7333. (Contributed by Alexander van der Vekens, 26-May-2017.)
((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ((𝐴𝐹𝐵)) = V)

Theoremaovvdm 44158 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 44159 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 44160 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 44161 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7194. (Contributed by Alexander van der Vekens, 26-May-2017.)
Rel dom 𝐹       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)

Theoremaovrcl 44162 Reverse closure for an operation value, analogous to afvvv 44118. In contrast to ovrcl 7197, 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 44163 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 44164 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 44165 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 44166 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 44167 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 44168 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 44169 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 44170* A frequently used special case of rspc2ev 3555 for operation values, analogous to rspceov 7203. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Theoremfnotaovb 44171 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6712. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Theoremffnaov 44172* An operation maps to a class to which all values belong, analogous to ffnov 7279. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

Theoremfaovcl 44173 Closure law for an operation, analogous to fovcl 7280. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)

Theoremaovmpt4g 44174* Value of a function given by the maps-to notation, analogous to ovmpt4g 7298. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)

Theoremaoprssdm 44175* Domain of closure of an operation. In contrast to oprssdm 7331, 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 44176 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7335 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 44177 Reverse closure law, in contrast to ndmovrcl 7336 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 44178 Any operation is commutative outside its domain, analogous to ndmovcom 7337. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )

Theoremndmaovass 44179 Any operation is associative outside its domain. In contrast to ndmovass 7338 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 44180 Any operation is distributive outside its domain. In contrast to ndmovdistr 7339 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 44093.

The current definition of the value (𝐹𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6348) 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 6693 and fvprc 6655). "(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 44092. 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 6694).

To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 44182) 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 44184). 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 44197, 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 44185). 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 44233), but (𝐹𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 44238.

The alternate definition, however, corresponds to the current definition ((𝐹𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 44234).

With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 44200.

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 6348 of (𝐹𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6662-> afv2eq1 44189, fveq2 6663-> afv2eq2 44190, nffv 6673-> nfafv2 44191, csbfv12 6706-> csbafv212g , rlimdm 14969-> rlimdmafv2 44231, tz6.12-1 6685-> tz6.12-1-afv2 44214, fveu 6653-> afv2eu 44211.

Six theorems proved by directly using df-fv 6348 are within a mathbox (fvsb 41564, uncov 35352) or not used (rlimdmafv 44150, avril1 28360) or experimental (dfafv2 44105, dfafv22 44232).

However, the remaining 11 theorems proved by directly using df-fv 6348 are used more or less often:

* fvex 6676: 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 44186 resp. afv2ex 44187). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6676.

* fvres 6682: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 44212). 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 44212 can be used instead of fvres 6682.

* tz6.12-2 6652 (-> tz6.12-2-afv2 44210): root theorem of many theorems which have not a strict analogue, and which are used many times:

** fvprc 6655 (-> afv2prc 44199), used in 193 proofs,

** tz6.12i 6689 (-> tz6.12i-afv2 44216), used - indirectly via fvbr0 6690 and fvrn0 6691 - in 19 proofs, and in fvclss 6999 used in fvclex 7670 used in fvresex 7671 (which is not used!) and in dcomex 9920 (used in 4 proofs),

** ndmfv 6693 (-> ndmafv2nrn ), used in 124 proofs

** nfunsn 6700 (-> nfunsnafv2 ), used by fvfundmfvn0 6701 (used in 3 proofs), and dffv2 6752 (not used)

** funpartfv 33830, setrec2lem1 45708 (mathboxes)

* fv2 6658: only used by elfv 6661, which is only used by fv3 6681, which is not used.

* dffv3 6659 (-> dfafv23 ): used by dffv4 6660 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 42013), by shftval 14494 (itself used in 11 proofs), by dffv5 33809 (mathbox) and by fvco2 6754 (-> afv2co2 44230).

* fvopab5 6796: used only by ajval 28756 (not used) and by adjval 29785, which is used in adjval2 29786 (not used) and in adjbdln 29978 (used in 7 proofs).

* zsum 15136: used (via isum 15137, sum0 15139, sumss 15142 and fsumsers 15146) in 76 proofs.

* isumshft 15255: used in pserdv2 25137 (used in logtayl 25363, binomcxplemdvsum 41467) , eftlub 15523 (used in 4 proofs), binomcxplemnotnn0 41468 (used in binomcxp 41469 only) and logtayl 25363 (used in 4 proofs).

* ovtpos 7923: used in 16 proofs.

* zprod 15352: used in 3 proofs: iprod 15353, zprodn0 15354 and prodss 15362

* iprodclim3 15415: 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 6658, dffv3 6659, fvopab5 6796, zsum 15136, isumshft 15255, ovtpos 7923 and zprod 15352 are not critical or are, hopefully, also valid for the alternative definition, fvex 6676, fvres 6682 and tz6.12-2 6652 (and the theorems based on them) are essential for the current definition of function values.

Syntaxcafv2 44181 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 44090.
class (𝐹''''𝐴)

Definitiondf-afv2 44182* 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 6348, and especially ndmfv 6693), (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 44183* 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 44184 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 44185 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 44186 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 44187 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 44188 Equality deduction for function value, analogous to fveq12d 6670. (Contributed by AV, 4-Sep-2022.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Theoremafv2eq1 44189 Equality theorem for function value, analogous to fveq1 6662. (Contributed by AV, 4-Sep-2022.)
(𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))

Theoremafv2eq2 44190 Equality theorem for function value, analogous to fveq2 6663. (Contributed by AV, 4-Sep-2022.)
(𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))

Theoremnfafv2 44191 Bound-variable hypothesis builder for function value, analogous to nffv 6673. To prove a deduction version of this analogous to nffvd 6675 is not easily possible because a deduction version of nfdfat 44100 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹''''𝐴)

Theoremcsbafv212g 44192 Move class substitution in and out of a function value, analogous to csbfv12 6706, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7198. (Contributed by AV, 4-Sep-2022.)
(𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))

Theoremfexafv2ex 44193 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 44194 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 44195 The value of a class outside its domain is not in the range, compare with ndmfv 6693. (Contributed by AV, 2-Sep-2022.)
𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremfunressndmafv2rn 44196 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 44197 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 44198 If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6700. (Contributed by AV, 2-Sep-2022.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremafv2prc 44199 A function's value at a proper class is not defined, compare with fvprc 6655. (Contributed by AV, 5-Sep-2022.)
𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremdfatafv2rnb 44200 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)

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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45818
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