![]() |
Metamath
Proof Explorer Theorem List (p. 472 of 491) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30946) |
![]() (30947-32469) |
![]() (32470-49035) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | afvfv0bi 47101 | 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.) |
⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V)) | ||
Theorem | afveu 47102* | The value of a function at a unique point, analogous to fveu 6895. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | ||
Theorem | fnbrafvb 47103 | Equivalence of function value and binary relation, analogous to fnbrfvb 6959. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | ||
Theorem | fnopafvb 47104 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6960. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹)) | ||
Theorem | funbrafvb 47105 | Equivalence of function value and binary relation, analogous to funbrfvb 6961. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
Theorem | funopafvb 47106 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6962. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
Theorem | funbrafv 47107 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6957. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)) | ||
Theorem | funbrafv2b 47108 | Function value in terms of a binary relation, analogous to funbrfv2b 6965. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵))) | ||
Theorem | dfafn5a 47109* | Representation of a function in terms of its values, analogous to dffn5 6966 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))) | ||
Theorem | dfafn5b 47110* | Representation of a function in terms of its values, analogous to dffn5 6966 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))) | ||
Theorem | fnrnafv 47111* | The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6967. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}) | ||
Theorem | afvelrnb 47112* | A member of a function's range is a value of the function, analogous to fvelrnb 6968 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵)) | ||
Theorem | afvelrnb0 47113* | A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6968. (Contributed by Alexander van der Vekens, 1-Jun-2017.) |
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵)) | ||
Theorem | dfaimafn 47114* | Alternate definition of the image of a function, analogous to dfimafn 6970. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) | ||
Theorem | dfaimafn2 47115* | Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6971. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)}) | ||
Theorem | afvelima 47116* | Function value in an image, analogous to fvelima 6973. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹'''𝑥) = 𝐴) | ||
Theorem | afvelrn 47117 | A function's value belongs to its range, analogous to fvelrn 7095. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹) | ||
Theorem | fnafvelrn 47118 | A function's value belongs to its range, analogous to fnfvelrn 7099. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹) | ||
Theorem | fafvelcdm 47119 | A function's value belongs to its codomain, analogous to ffvelcdm 7100. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵) | ||
Theorem | ffnafv 47120* | A function maps to a class to which all values belong, analogous to ffnfv 7138. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)) | ||
Theorem | afvres 47121 | The value of a restricted function, analogous to fvres 6925. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴)) | ||
Theorem | tz6.12-afv 47122* | Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6931. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹'''𝐴) = 𝑦) | ||
Theorem | tz6.12-1-afv 47123* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6929. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦) | ||
Theorem | dmfcoafv 47124 | Domains of a function composition, analogous to dmfco 7004. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹)) | ||
Theorem | afvco2 47125 | Value of a function composition, analogous to fvco2 7005. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋))) | ||
Theorem | rlimdmafv 47126 | Two ways to express that a function has a limit, analogous to rlimdm 15583. (Contributed by Alexander van der Vekens, 27-Nov-2017.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 '''𝐹))) | ||
Theorem | aoveq123d 47127 | Equality deduction for operation value, analogous to oveq123d 7451. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) ) | ||
Theorem | nfaov 47128 | Bound-variable hypothesis builder for operation value, analogous to nfov 7460. To prove a deduction version of this analogous to nfovd 7459 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 47085). (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵)) | ||
Theorem | csbaovg 47129 | Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌ ((𝐵𝐹𝐶)) = ((⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶)) ) | ||
Theorem | aovfundmoveq 47130 | 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)) | ||
Theorem | aovnfundmuv 47131 | 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) | ||
Theorem | ndmaov 47132 | The value of an operation outside its domain, analogous to ndmafv 47089. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | ||
Theorem | ndmaovg 47133 | The value of an operation outside its domain, analogous to ndmovg 7615. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V) | ||
Theorem | aovvdm 47134 | 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) | ||
Theorem | nfunsnaov 47135 | 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) | ||
Theorem | aovvfunressn 47136 | 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〉})) | ||
Theorem | aovprc 47137 | The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7468. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ Rel dom 𝐹 ⇒ ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V) | ||
Theorem | aovrcl 47138 | Reverse closure for an operation value, analogous to afvvv 47094. In contrast to ovrcl 7471, 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)) | ||
Theorem | aovpcov0 47139 | 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) = ∅) | ||
Theorem | aovnuoveq 47140 | 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)) | ||
Theorem | aovvoveq 47141 | 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)) | ||
Theorem | aov0ov0 47142 | 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) = ∅) | ||
Theorem | aovovn0oveq 47143 | 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)) | ||
Theorem | aov0nbovbi 47144 | 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)) | ||
Theorem | aovov0bi 47145 | 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)) | ||
Theorem | rspceaov 47146* | A frequently used special case of rspc2ev 3634 for operation values, analogous to rspceov 7479. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) ) | ||
Theorem | fnotaovb 47147 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6960. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) | ||
Theorem | ffnaov 47148* | An operation maps to a class to which all values belong, analogous to ffnov 7558. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶)) | ||
Theorem | faovcl 47149 | Closure law for an operation, analogous to fovcl 7560. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶) | ||
Theorem | aovmpt4g 47150* | Value of a function given by the maps-to notation, analogous to ovmpt4g 7579. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶) | ||
Theorem | aoprssdm 47151* | Domain of closure of an operation. In contrast to oprssdm 7613, no additional property for S (¬ ∅ ∈ 𝑆) is required! (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆) ⇒ ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 | ||
Theorem | ndmaovcl 47152 | The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7617 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) & ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) & ⊢ ((𝐴𝐹𝐵)) ∈ V ⇒ ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆 | ||
Theorem | ndmaovrcl 47153 | Reverse closure law, in contrast to ndmovrcl 7618 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 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | ||
Theorem | ndmaovcom 47154 | Any operation is commutative outside its domain, analogous to ndmovcom 7619. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) ) | ||
Theorem | ndmaovass 47155 | Any operation is associative outside its domain. In contrast to ndmovass 7620 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) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) ) | ||
Theorem | ndmaovdistr 47156 | Any operation is distributive outside its domain. In contrast to ndmovdistr 7621 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) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐺 ((𝐵𝐹𝐶)) )) = (( ((𝐴𝐺𝐵)) 𝐹 ((𝐴𝐺𝐶)) )) ) | ||
In the following, a second approach is followed to define function values alternately to df-afv 47069. The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6570) assures that this value is always a set, see fex 7245. 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 6941 and fvprc 6898). "(𝐹‘𝐴) 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 47068. 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 (𝐹‘𝐴) = ∅ alone it cannot be decided/derived whether (𝐹‘𝐴) 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 𝐹 ∧ 𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6942). To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 47158) 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 47160). 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 47173, 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 47161). 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 → (𝐹‘𝐴) = ∅ (see afv2ndeffv0 47209), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 47214. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 47210). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 47176. An interesting question would be if (𝐹‘𝐴) 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 6570 of (𝐹‘𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6905-> afv2eq1 47165, fveq2 6906-> afv2eq2 47166, nffv 6916-> nfafv2 47167, csbfv12 6954-> csbafv212g , rlimdm 15583-> rlimdmafv2 47207, tz6.12-1 6929-> tz6.12-1-afv2 47190, fveu 6895-> afv2eu 47187. Six theorems proved by directly using df-fv 6570 are within a mathbox (fvsb 44447, uncov 37587) or not used (rlimdmafv 47126, avril1 30491) or experimental (dfafv2 47081, dfafv22 47208). However, the remaining 11 theorems proved by directly using df-fv 6570 are used more or less often: * fvex 6919: 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 47162 resp. afv2ex 47163). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6919. * fvres 6925: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 47188). In the undefined case such a theorem cannot exist (without additional assumptions), because the range of (𝐹 ↾ 𝐵) 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 47188 can be used instead of fvres 6925. * tz6.12-2 6894 (-> tz6.12-2-afv2 47186): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6898 (-> afv2prc 47175), used in 193 proofs, ** tz6.12i 6934 (-> tz6.12i-afv2 47192), used - indirectly via fvbr0 6935 and fvrn0 6936 - in 19 proofs, and in fvclss 7260 used in fvclex 7981 used in fvresex 7982 (which is not used!) and in dcomex 10484 (used in 4 proofs), ** ndmfv 6941 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6948 (-> nfunsnafv2 ), used by fvfundmfvn0 6949 (used in 3 proofs), and dffv2 7003 (not used) ** funpartfv 35926, setrec2lem1 48923 (mathboxes) * fv2 6901: only used by elfv 6904, which is only used by fv3 6924, which is not used. * dffv3 6902 (-> dfafv23 ): used by dffv4 6903 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 44896), by shftval 15109 (itself used in 11 proofs), by dffv5 35905 (mathbox) and by fvco2 7005 (-> afv2co2 47206). * fvopab5 7048: used only by ajval 30889 (not used) and by adjval 31918, which is used in adjval2 31919 (not used) and in adjbdln 32111 (used in 7 proofs). * zsum 15750: used (via isum 15751, sum0 15753, sumss 15756 and fsumsers 15760) in 76 proofs. * isumshft 15871: used in pserdv2 26488 (used in logtayl 26716, binomcxplemdvsum 44350) , eftlub 16141 (used in 4 proofs), binomcxplemnotnn0 44351 (used in binomcxp 44352 only) and logtayl 26716 (used in 4 proofs). * ovtpos 8264: used in 16 proofs. * zprod 15969: used in 3 proofs: iprod 15970, zprodn0 15971 and prodss 15979 * iprodclim3 16032: 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 6901, dffv3 6902, fvopab5 7048, zsum 15750, isumshft 15871, ovtpos 8264 and zprod 15969 are not critical or are, hopefully, also valid for the alternative definition, fvex 6919, fvres 6925 and tz6.12-2 6894 (and the theorems based on them) are essential for the current definition of function values. | ||
Syntax | cafv2 47157 | 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 47066. |
class (𝐹''''𝐴) | ||
Definition | df-afv2 47158* | 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 (𝐹‘𝐴) = ∅ (see comment of df-fv 6570, and especially ndmfv 6941), (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) | ||
Theorem | dfatafv2iota 47159* | 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 (𝐹‘𝐴)). (Contributed by AV, 2-Sep-2022.) |
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | ||
Theorem | ndfatafv2 47160 | 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) | ||
Theorem | ndfatafv2undef 47161 | 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 𝐹 ∈ 𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹)) | ||
Theorem | dfatafv2ex 47162 | 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) | ||
Theorem | afv2ex 47163 | The alternate function value is always a set if the range of the function is a set. (Contributed by AV, 2-Sep-2022.) |
⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) | ||
Theorem | afv2eq12d 47164 | Equality deduction for function value, analogous to fveq12d 6913. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵)) | ||
Theorem | afv2eq1 47165 | Equality theorem for function value, analogous to fveq1 6905. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴)) | ||
Theorem | afv2eq2 47166 | Equality theorem for function value, analogous to fveq2 6906. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵)) | ||
Theorem | nfafv2 47167 | Bound-variable hypothesis builder for function value, analogous to nffv 6916. To prove a deduction version of this analogous to nffvd 6918 is not easily possible because a deduction version of nfdfat 47076 cannot be shown easily. (Contributed by AV, 4-Sep-2022.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹''''𝐴) | ||
Theorem | csbafv212g 47168 | Move class substitution in and out of a function value, analogous to csbfv12 6954, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7474. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) | ||
Theorem | fexafv2ex 47169 | 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.) |
⊢ (𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) | ||
Theorem | ndfatafv2nrn 47170 | 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) | ||
Theorem | ndmafv2nrn 47171 | The value of a class outside its domain is not in the range, compare with ndmfv 6941. (Contributed by AV, 2-Sep-2022.) |
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹) | ||
Theorem | funressndmafv2rn 47172 | 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) | ||
Theorem | afv2ndefb 47173 | 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) | ||
Theorem | nfunsnafv2 47174 | If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6948. (Contributed by AV, 2-Sep-2022.) |
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹) | ||
Theorem | afv2prc 47175 | A function's value at a proper class is not defined, compare with fvprc 6898. (Contributed by AV, 5-Sep-2022.) |
⊢ (¬ 𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹) | ||
Theorem | dfatafv2rnb 47176 | 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) | ||
Theorem | afv2orxorb 47177 | 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))) | ||
Theorem | dmafv2rnb 47178 | 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)) | ||
Theorem | fundmafv2rnb 47179 | 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)) | ||
Theorem | afv2elrn 47180 | An alternate function value belongs to the range of the function, analogous to fvelrn 7095. (Contributed by AV, 3-Sep-2022.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹) | ||
Theorem | afv20defat 47181 | 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) | ||
Theorem | fnafv2elrn 47182 | An alternate function value belongs to the range of the function, analogous to fnfvelrn 7099. (Contributed by AV, 2-Sep-2022.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹) | ||
Theorem | fafv2elcdm 47183 | An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7100. (Contributed by AV, 2-Sep-2022.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵) | ||
Theorem | fafv2elrnb 47184 | 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:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹)) | ||
Theorem | fcdmvafv2v 47185 | 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:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V) | ||
Theorem | tz6.12-2-afv2 47186* | Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6894. (Contributed by AV, 5-Sep-2022.) |
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹) | ||
Theorem | afv2eu 47187* | The value of a function at a unique point, analogous to fveu 6895. (Contributed by AV, 5-Sep-2022.) |
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | ||
Theorem | afv2res 47188 | The value of a restricted function for an argument at which the function is defined. Analog to fvres 6925. (Contributed by AV, 5-Sep-2022.) |
⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴)) | ||
Theorem | tz6.12-afv2 47189* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6931. (Contributed by AV, 5-Sep-2022.) |
⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹''''𝐴) = 𝑦) | ||
Theorem | tz6.12-1-afv2 47190* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6929. (Contributed by AV, 5-Sep-2022.) |
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦) | ||
Theorem | tz6.12c-afv2 47191* | Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6928. (Contributed by AV, 5-Sep-2022.) |
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) | ||
Theorem | tz6.12i-afv2 47192 | Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6934. (Contributed by AV, 5-Sep-2022.) |
⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵)) | ||
Theorem | funressnbrafv2 47193 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6957. (Contributed by AV, 7-Sep-2022.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵)) | ||
Theorem | dfatbrafv2b 47194 | Equivalence of function value and binary relation, analogous to fnbrfvb 6959 or funbrfvb 6961. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 47162). (Contributed by AV, 6-Sep-2022.) |
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
Theorem | dfatopafv2b 47195 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6960 or funopfvb 6962. (Contributed by AV, 6-Sep-2022.) |
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
Theorem | funbrafv2 47196 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6957. (Contributed by AV, 6-Sep-2022.) |
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵)) | ||
Theorem | fnbrafv2b 47197 | Equivalence of function value and binary relation, analogous to fnbrfvb 6959. (Contributed by AV, 6-Sep-2022.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | ||
Theorem | fnopafv2b 47198 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6960. (Contributed by AV, 6-Sep-2022.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹)) | ||
Theorem | funbrafv22b 47199 | Equivalence of function value and binary relation, analogous to funbrfvb 6961. (Contributed by AV, 6-Sep-2022.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
Theorem | funopafv2b 47200 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6962. (Contributed by AV, 6-Sep-2022.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |