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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | afvpcfv0 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 → (𝐹‘𝐴) = ∅) | ||
Theorem | afvnufveq 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) = (𝐹‘𝐴)) | ||
Theorem | afvvfveq 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) = (𝐹‘𝐴)) | ||
Theorem | afv0fv0 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) = ∅ → (𝐹‘𝐴) = ∅) | ||
Theorem | afvfvn0fveq 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.) |
⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹‘𝐴)) | ||
Theorem | afv0nbfvbi 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 ↔ (𝐹‘𝐴) ∈ 𝐵)) | ||
Theorem | afvfv0bi 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.) |
⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V)) | ||
Theorem | afveu 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) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | ||
Theorem | fnbrafvb 42209 | Equivalence of function value and binary relation, analogous to fnbrfvb 6497. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | ||
Theorem | fnopafvb 42210 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6498. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹)) | ||
Theorem | funbrafvb 42211 | Equivalence of function value and binary relation, analogous to funbrfvb 6499. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
Theorem | funopafvb 42212 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6500. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
Theorem | funbrafv 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)) | ||
Theorem | funbrafv2b 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))) | ||
Theorem | dfafn5a 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 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))) | ||
Theorem | dfafn5b 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.) |
⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))) | ||
Theorem | fnrnafv 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 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}) | ||
Theorem | afvelrnb 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 𝐴 ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵)) | ||
Theorem | afvelrnb0 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 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵)) | ||
Theorem | dfaimafn 42220* | Alternate definition of the image of a function, analogous to dfimafn 6507. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) | ||
Theorem | dfaimafn2 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 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)}) | ||
Theorem | afvelima 42222* | Function value in an image, analogous to fvelima 6510. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹'''𝑥) = 𝐴) | ||
Theorem | afvelrn 42223 | A function's value belongs to its range, analogous to fvelrn 6618. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹) | ||
Theorem | fnafvelrn 42224 | A function's value belongs to its range, analogous to fnfvelrn 6622. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹) | ||
Theorem | fafvelrn 42225 | A function's value belongs to its codomain, analogous to ffvelrn 6623. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵) | ||
Theorem | ffnafv 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:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)) | ||
Theorem | afvres 42227 | The value of a restricted function, analogous to fvres 6467. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴)) | ||
Theorem | tz6.12-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 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹'''𝐴) = 𝑦) | ||
Theorem | tz6.12-1-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) | ||
Theorem | dmfcoafv 42230 | Domains of a function composition, analogous to dmfco 6534. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹)) | ||
Theorem | afvco2 42231 | Value of a function composition, analogous to fvco2 6535. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋))) | ||
Theorem | rlimdmafv 42232 | Two ways to express that a function has a limit, analogous to rlimdm 14699. (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 42233 | Equality deduction for operation value, analogous to oveq123d 6945. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) ) | ||
Theorem | nfaov 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.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵)) | ||
Theorem | csbaovg 42235 | 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 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)) | ||
Theorem | aovnfundmuv 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) | ||
Theorem | ndmaov 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) | ||
Theorem | ndmaovg 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) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V) | ||
Theorem | aovvdm 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) | ||
Theorem | nfunsnaov 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) | ||
Theorem | aovvfunressn 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〉})) | ||
Theorem | aovprc 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) | ||
Theorem | aovrcl 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)) | ||
Theorem | aovpcov0 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) = ∅) | ||
Theorem | aovnuoveq 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)) | ||
Theorem | aovvoveq 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)) | ||
Theorem | aov0ov0 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) = ∅) | ||
Theorem | aovovn0oveq 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)) | ||
Theorem | aov0nbovbi 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)) | ||
Theorem | aovov0bi 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)) | ||
Theorem | rspceaov 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.) |
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) ) | ||
Theorem | fnotaovb 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) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ( ((𝐶𝐹𝐷)) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) | ||
Theorem | ffnaov 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) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥𝐹𝑦)) ∈ 𝐶)) | ||
Theorem | faovcl 42255 | Closure law for an operation, analogous to fovcl 7044. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶) | ||
Theorem | aovmpt4g 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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶) | ||
Theorem | aoprssdm 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.) |
⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦)) ∈ 𝑆) ⇒ ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 | ||
Theorem | ndmaovcl 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) & ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆) & ⊢ ((𝐴𝐹𝐵)) ∈ V ⇒ ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆 | ||
Theorem | ndmaovrcl 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 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | ||
Theorem | ndmaovcom 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) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) ) | ||
Theorem | ndmaovass 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) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) ) | ||
Theorem | ndmaovdistr 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) ⇒ ⊢ (¬ (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 42175. The current definition of the value (𝐹‘𝐴) 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). "(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 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 (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 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 → (𝐹‘𝐴) = ∅ (see afv2ndeffv0 42315), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 42320. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (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 (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 6145 of (𝐹‘𝐴), 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 (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 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. | ||
Syntax | cafv2 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) | ||
Definition | df-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 (𝐹‘𝐴) = ∅ (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) | ||
Theorem | dfatafv2iota 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 (𝐹‘𝐴)). (Contributed by AV, 2-Sep-2022.) |
⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥)) | ||
Theorem | ndfatafv2 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) | ||
Theorem | ndfatafv2undef 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 𝐹 ∈ 𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹)) | ||
Theorem | dfatafv2ex 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) | ||
Theorem | afv2ex 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 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) | ||
Theorem | afv2eq12d 42270 | Equality deduction for function value, analogous to fveq12d 6455. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵)) | ||
Theorem | afv2eq1 42271 | Equality theorem for function value, analogous to fveq1 6447. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴)) | ||
Theorem | afv2eq2 42272 | Equality theorem for function value, analogous to fveq2 6448. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵)) | ||
Theorem | nfafv2 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.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹''''𝐴) | ||
Theorem | csbafv212g 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.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) | ||
Theorem | fexafv2ex 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.) |
⊢ (𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V) | ||
Theorem | ndfatafv2nrn 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) | ||
Theorem | ndmafv2nrn 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) | ||
Theorem | funressndmafv2rn 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) | ||
Theorem | afv2ndefb 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) | ||
Theorem | nfunsnafv2 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) | ||
Theorem | afv2prc 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) | ||
Theorem | dfatafv2rnb 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) | ||
Theorem | afv2orxorb 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))) | ||
Theorem | dmafv2rnb 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)) | ||
Theorem | fundmafv2rnb 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)) | ||
Theorem | afv2elrn 42286 | An alternate function value belongs to the range of the function, analogous to fvelrn 6618. (Contributed by AV, 3-Sep-2022.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹) | ||
Theorem | afv20defat 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) | ||
Theorem | fnafv2elrn 42288 | An alternate function value belongs to the range of the function, analogous to fnfvelrn 6622. (Contributed by AV, 2-Sep-2022.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹) | ||
Theorem | fafv2elrn 42289 | An alternate function value belongs to the codomain of the function, analogous to ffvelrn 6623. (Contributed by AV, 2-Sep-2022.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵) | ||
Theorem | fafv2elrnb 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:𝐴⟶𝐵 → (𝐶 ∈ 𝐴 ↔ (𝐹''''𝐶) ∈ ran 𝐹)) | ||
Theorem | frnvafv2v 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:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V) | ||
Theorem | tz6.12-2-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) | ||
Theorem | afv2eu 42293* | The value of a function at a unique point, analogous to fveu 6439. (Contributed by AV, 5-Sep-2022.) |
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | ||
Theorem | afv2res 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 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴)) | ||
Theorem | tz6.12-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 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹''''𝐴) = 𝑦) | ||
Theorem | tz6.12-1-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) | ||
Theorem | tz6.12c-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) = 𝑦 ↔ 𝐴𝐹𝑦)) | ||
Theorem | tz6.12i-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) = 𝐵 → 𝐴𝐹𝐵)) | ||
Theorem | funressnbrafv2 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.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵)) | ||
Theorem | dfatbrafv2b 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 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) |
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