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Theorem List for Metamath Proof Explorer - 42201-42300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremafvpcfv0 42201 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 42202 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 42203 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 42204 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 42205 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 42206 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 42207 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 42208* The value of a function at a unique point, analogous to fveu 6439. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})

Theoremfnbrafvb 42209 Equivalence of function value and binary relation, analogous to fnbrfvb 6497. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))

Theoremfnopafvb 42210 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6498. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Theoremfunbrafvb 42211 Equivalence of function value and binary relation, analogous to funbrfvb 6499. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunopafvb 42212 Equivalence of function value and ordered pair membership, analogous to funopfvb 6500. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

Theoremfunbrafv 42213 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6495. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Theoremfunbrafv2b 42214 Function value in terms of a binary relation, analogous to funbrfv2b 6502. (Contributed by Alexander van der Vekens, 25-May-2017.)
(Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵)))

Theoremdfafn5a 42215* Representation of a function in terms of its values, analogous to dffn5 6503 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))

Theoremdfafn5b 42216* Representation of a function in terms of its values, analogous to dffn5 6503 (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 42217* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6504. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})

Theoremafvelrnb 42218* A member of a function's range is a value of the function, analogous to fvelrnb 6505 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 42219* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6505. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
(𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))

Theoremdfaimafn 42220* Alternate definition of the image of a function, analogous to dfimafn 6507. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})

Theoremdfaimafn2 42221* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6508. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})

Theoremafvelima 42222* Function value in an image, analogous to fvelima 6510. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹'''𝑥) = 𝐴)

Theoremafvelrn 42223 A function's value belongs to its range, analogous to fvelrn 6618. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)

Theoremfnafvelrn 42224 A function's value belongs to its range, analogous to fnfvelrn 6622. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Theoremfafvelrn 42225 A function's value belongs to its codomain, analogous to ffvelrn 6623. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹'''𝐶) ∈ 𝐵)

Theoremffnafv 42226* A function maps to a class to which all values belong, analogous to ffnfv 6654. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))

Theoremafvres 42227 The value of a restricted function, analogous to fvres 6467. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))

Theoremtz6.12-afv 42228* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6471. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹'''𝐴) = 𝑦)

Theoremtz6.12-1-afv 42229* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6470. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦)

Theoremdmfcoafv 42230 Domains of a function composition, analogous to dmfco 6534. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹))

Theoremafvco2 42231 Value of a function composition, analogous to fvco2 6535. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Theoremrlimdmafv 42232 Two ways to express that a function has a limit, analogous to rlimdm 14699. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → sup(𝐴, ℝ*, < ) = +∞)       (𝜑 → (𝐹 ∈ dom ⇝𝑟𝐹𝑟 ( ⇝𝑟 '''𝐹)))

20.36.4.6  Alternative definition of the value of an operation

Theoremaoveq123d 42233 Equality deduction for operation value, analogous to oveq123d 6945. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) )

Theoremnfaov 42234 Bound-variable hypothesis builder for operation value, analogous to nfov 6954. To prove a deduction version of this analogous to nfovd 6953 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 42191). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐵       𝑥 ((𝐴𝐹𝐵))

Theoremcsbaovg 42235 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 42236 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 42237 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 42238 The value of an operation outside its domain, analogous to ndmafv 42195. (Contributed by Alexander van der Vekens, 26-May-2017.)
(¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)

Theoremndmaovg 42239 The value of an operation outside its domain, analogous to ndmovg 7096. (Contributed by Alexander van der Vekens, 26-May-2017.)
((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ((𝐴𝐹𝐵)) = V)

Theoremaovvdm 42240 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 42241 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 42242 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 42243 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 6961. (Contributed by Alexander van der Vekens, 26-May-2017.)
Rel dom 𝐹       (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V)

Theoremaovrcl 42244 Reverse closure for an operation value, analogous to afvvv 42200. In contrast to ovrcl 6964, 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 42245 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 42246 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 42247 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 42248 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 42249 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 42250 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 42251 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 42252* A frequently used special case of rspc2ev 3526 for operation values, analogous to rspceov 6970. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )

Theoremfnotaovb 42253 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6498. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Theoremffnaov 42254* An operation maps to a class to which all values belong, analogous to ffnov 7043. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶))

Theoremfaovcl 42255 Closure law for an operation, analogous to fovcl 7044. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹:(𝑅 × 𝑆)⟶𝐶       ((𝐴𝑅𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶)

Theoremaovmpt4g 42256* Value of a function given by the maps-to notation, analogous to ovmpt4g 7062. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)

Theoremaoprssdm 42257* Domain of closure of an operation. In contrast to oprssdm 7094, 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 42258 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7098 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 42259 Reverse closure law, in contrast to ndmovrcl 7099 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 42260 Any operation is commutative outside its domain, analogous to ndmovcom 7100. (Contributed by Alexander van der Vekens, 26-May-2017.)
dom 𝐹 = (𝑆 × 𝑆)       (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )

Theoremndmaovass 42261 Any operation is associative outside its domain. In contrast to ndmovass 7101 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 42262 Any operation is distributive outside its domain. In contrast to ndmovdistr 7102 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)) )) )

20.36.5  Alternative definitions of function values (2)

In the following, a second approach is followed to define function values alternately to df-afv 42175.

The current definition of the value (𝐹𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6145) assures that this value is always a set, see fex 6763. 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 6478 and fvprc 6441). "(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 42174. 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 6479).

To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 42264) 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 42266). 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 42279, 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 42267). 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 42315), but (𝐹𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 42320.

The alternate definition, however, corresponds to the current definition ((𝐹𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 42316).

With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 42282.

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 6145 of (𝐹𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6447-> afv2eq1 42271, fveq2 6448-> afv2eq2 42272, nffv 6458-> nfafv2 42273, csbfv12 6492-> csbafv212g , rlimdm 14699-> rlimdmafv2 42313, tz6.12-1 6470-> tz6.12-1-afv2 42296, fveu 6439-> afv2eu 42293.

Six theorems proved by directly using df-fv 6145 are within a mathbox (fvsb 39624, uncov 34024) or not used (rlimdmafv 42232, avril1 27911) or experimental (dfafv2 42187, dfafv22 42314).

However, the remaining 11 theorems proved by directly using df-fv 6145 are used more or less often:

* fvex 6461: 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 42268 resp. afv2ex 42269). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6461.

* fvres 6467: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 42294). 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 42294 can be used instead of fvres 6467.

* tz6.12-2 6438 (-> tz6.12-2-afv2 42292): root theorem of many theorems which have not a strict analogue, and which are used many times:

** fvprc 6441 (-> afv2prc 42281), used in 193 proofs,

** tz6.12i 6474 (-> tz6.12i-afv2 42298), used - indirectly via fvbr0 6475 and fvrn0 6476 - in 19 proofs, and in fvclss 6774 used in fvclex 7419 used in fvresex 7420 (which is not used!) and in dcomex 9606 (used in 4 proofs),

** ndmfv 6478 (-> ndmafv2nrn ), used in 124 proofs

** nfunsn 6486 (-> nfunsnafv2 ), used by fvfundmfvn0 6487 (used in 3 proofs), and dffv2 6533 (not used)

** funpartfv 32649, setrec2lem1 43559 (mathboxes)

* fv2 6443: only used by elfv 6446, which is only used by fv3 6466, which is not used.

* dffv3 6444 (-> dfafv23 ): used by dffv4 6445 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 40082), by shftval 14227 (itself used in 11 proofs), by dffv5 32628 (mathbox) and by fvco2 6535 (-> afv2co2 42312).

* fvopab5 6574: used only by ajval 28306 (not used) and by adjval 29338, which is used in adjval2 29339 (not used) and in adjbdln 29531 (used in 7 proofs).

* zsum 14865: used (via isum 14866, sum0 14868, sumss 14871 and fsumsers 14875) in 76 proofs.

* isumshft 14984: used in pserdv2 24632 (used in logtayl 24854, binomcxplemdvsum 39524) , eftlub 15250 (used in 4 proofs), binomcxplemnotnn0 39525 (used in binomcxp 39526 only) and logtayl 24854 (used in 4 proofs).

* ovtpos 7651: used in 16 proofs.

* zprod 15079: used in 3 proofs: iprod 15080, zprodn0 15081 and prodss 15089

* iprodclim3 15142: 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 6443, dffv3 6444, fvopab5 6574, zsum 14865, isumshft 14984, ovtpos 7651 and zprod 15079 are not critical or are, hopefully, also valid for the alternative definition, fvex 6461, fvres 6467 and tz6.12-2 6438 (and the theorems based on them) are essential for the current definition of function values.

Syntaxcafv2 42263 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 42172.
class (𝐹''''𝐴)

Definitiondf-afv2 42264* 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 6145, and especially ndmfv 6478), (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 42265* 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 42266 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 42267 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 42268 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 42269 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 42270 Equality deduction for function value, analogous to fveq12d 6455. (Contributed by AV, 4-Sep-2022.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Theoremafv2eq1 42271 Equality theorem for function value, analogous to fveq1 6447. (Contributed by AV, 4-Sep-2022.)
(𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴))

Theoremafv2eq2 42272 Equality theorem for function value, analogous to fveq2 6448. (Contributed by AV, 4-Sep-2022.)
(𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵))

Theoremnfafv2 42273 Bound-variable hypothesis builder for function value, analogous to nffv 6458. To prove a deduction version of this analogous to nffvd 6460 is not easily possible because a deduction version of nfdfat 42182 cannot be shown easily. (Contributed by AV, 4-Sep-2022.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹''''𝐴)

Theoremcsbafv212g 42274 Move class substitution in and out of a function value, analogous to csbfv12 6492, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6965. (Contributed by AV, 4-Sep-2022.)
(𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))

Theoremfexafv2ex 42275 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 42276 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 42277 The value of a class outside its domain is not in the range, compare with ndmfv 6478. (Contributed by AV, 2-Sep-2022.)
𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremfunressndmafv2rn 42278 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 42279 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 42280 If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6486. (Contributed by AV, 2-Sep-2022.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremafv2prc 42281 A function's value at a proper class is not defined, compare with fvprc 6441. (Contributed by AV, 5-Sep-2022.)
𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremdfatafv2rnb 42282 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 42283 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 42284 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))

Theoremfundmafv2rnb 42285 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 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))

Theoremafv2elrn 42286 An alternate function value belongs to the range of the function, analogous to fvelrn 6618. (Contributed by AV, 3-Sep-2022.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)

Theoremafv20defat 42287 If the alternate function value at an argument is the empty set, the function is defined at this argument. (Contributed by AV, 3-Sep-2022.)
((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)

Theoremfnafv2elrn 42288 An alternate function value belongs to the range of the function, analogous to fnfvelrn 6622. (Contributed by AV, 2-Sep-2022.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹''''𝐵) ∈ ran 𝐹)

Theoremfafv2elrn 42289 An alternate function value belongs to the codomain of the function, analogous to ffvelrn 6623. (Contributed by AV, 2-Sep-2022.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹''''𝐶) ∈ 𝐵)

Theoremfafv2elrnb 42290 An alternate function value is defined, i.e., belongs to the range of the function, iff its argument is in the domain of the function. (Contributed by AV, 3-Sep-2022.)
(𝐹:𝐴𝐵 → (𝐶𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹))

Theoremfrnvafv2v 42291 If the codomain of a function is a set, the alternate function value is always also a set. (Contributed by AV, 4-Sep-2022.)
((𝐹:𝐴𝐵𝐵𝑉) → (𝐹''''𝐶) ∈ V)

Theoremtz6.12-2-afv2 42292* Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6438. (Contributed by AV, 5-Sep-2022.)
(¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹)

Theoremafv2eu 42293* The value of a function at a unique point, analogous to fveu 6439. (Contributed by AV, 5-Sep-2022.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = {𝑥𝐴𝐹𝑥})

Theoremafv2res 42294 The value of a restricted function for an argument at which the function is defined. Analog to fvres 6467. (Contributed by AV, 5-Sep-2022.)
((𝐹 defAt 𝐴𝐴𝐵) → ((𝐹𝐵)''''𝐴) = (𝐹''''𝐴))

Theoremtz6.12-afv2 42295* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6471. (Contributed by AV, 5-Sep-2022.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹''''𝐴) = 𝑦)

Theoremtz6.12-1-afv2 42296* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6470. (Contributed by AV, 5-Sep-2022.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦)

Theoremtz6.12c-afv2 42297* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6473. (Contributed by AV, 5-Sep-2022.)
(∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))

Theoremtz6.12i-afv2 42298 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6474. (Contributed by AV, 5-Sep-2022.)
(𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunressnbrafv2 42299 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6495. (Contributed by AV, 7-Sep-2022.)
(((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Theoremdfatbrafv2b 42300 Equivalence of function value and binary relation, analogous to fnbrfvb 6497 or funbrfvb 6499. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 42268). (Contributed by AV, 6-Sep-2022.)
((𝐹 defAt 𝐴𝐵𝑊) → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))

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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 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