Home | Metamath
Proof Explorer Theorem List (p. 435 of 450) | < 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: | Metamath Proof Explorer
(1-28695) |
Hilbert Space Explorer
(28696-30218) |
Users' Mathboxes
(30219-44955) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | afveu 43401* | The value of a function at a unique point, analogous to fveu 6661. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | ||
Theorem | fnbrafvb 43402 | Equivalence of function value and binary relation, analogous to fnbrfvb 6718. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | ||
Theorem | fnopafvb 43403 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6719. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹)) | ||
Theorem | funbrafvb 43404 | Equivalence of function value and binary relation, analogous to funbrfvb 6720. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
Theorem | funopafvb 43405 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6721. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
Theorem | funbrafv 43406 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6716. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)) | ||
Theorem | funbrafv2b 43407 | Function value in terms of a binary relation, analogous to funbrfv2b 6723. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹'''𝐴) = 𝐵))) | ||
Theorem | dfafn5a 43408* | Representation of a function in terms of its values, analogous to dffn5 6724 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))) | ||
Theorem | dfafn5b 43409* | Representation of a function in terms of its values, analogous to dffn5 6724 (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 43410* | The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6725. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹'''𝑥)}) | ||
Theorem | afvelrnb 43411* | A member of a function's range is a value of the function, analogous to fvelrnb 6726 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 43412* | A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6726. (Contributed by Alexander van der Vekens, 1-Jun-2017.) |
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝐵)) | ||
Theorem | dfaimafn 43413* | Alternate definition of the image of a function, analogous to dfimafn 6728. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) | ||
Theorem | dfaimafn2 43414* | Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6729. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)}) | ||
Theorem | afvelima 43415* | Function value in an image, analogous to fvelima 6731. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹'''𝑥) = 𝐴) | ||
Theorem | afvelrn 43416 | A function's value belongs to its range, analogous to fvelrn 6844. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹) | ||
Theorem | fnafvelrn 43417 | A function's value belongs to its range, analogous to fnfvelrn 6848. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) ∈ ran 𝐹) | ||
Theorem | fafvelrn 43418 | A function's value belongs to its codomain, analogous to ffvelrn 6849. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹'''𝐶) ∈ 𝐵) | ||
Theorem | ffnafv 43419* | A function maps to a class to which all values belong, analogous to ffnfv 6882. (Contributed by Alexander van der Vekens, 25-May-2017.) |
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)) | ||
Theorem | afvres 43420 | The value of a restricted function, analogous to fvres 6689. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴)) | ||
Theorem | tz6.12-afv 43421* | Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6693. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹'''𝐴) = 𝑦) | ||
Theorem | tz6.12-1-afv 43422* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6692. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹'''𝐴) = 𝑦) | ||
Theorem | dmfcoafv 43423 | Domains of a function composition, analogous to dmfco 6757. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺'''𝐴) ∈ dom 𝐹)) | ||
Theorem | afvco2 43424 | Value of a function composition, analogous to fvco2 6758. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋))) | ||
Theorem | rlimdmafv 43425 | Two ways to express that a function has a limit, analogous to rlimdm 14908. (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 43426 | Equality deduction for operation value, analogous to oveq123d 7177. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐶)) = ((𝐵𝐺𝐷)) ) | ||
Theorem | nfaov 43427 | Bound-variable hypothesis builder for operation value, analogous to nfov 7186. To prove a deduction version of this analogous to nfovd 7185 is not quickly possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of alternative operation values is based on are not available (see nfafv 43384). (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 ((𝐴𝐹𝐵)) | ||
Theorem | csbaovg 43428 | 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 43429 | 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 43430 | 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 43431 | The value of an operation outside its domain, analogous to ndmafv 43388. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | ||
Theorem | ndmaovg 43432 | The value of an operation outside its domain, analogous to ndmovg 7331. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V) | ||
Theorem | aovvdm 43433 | 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 43434 | 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 43435 | 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 43436 | The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7194. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ Rel dom 𝐹 ⇒ ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V) | ||
Theorem | aovrcl 43437 | Reverse closure for an operation value, analogous to afvvv 43393. In contrast to ovrcl 7197, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ Rel dom 𝐹 ⇒ ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | aovpcov0 43438 | 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 43439 | 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 43440 | 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 43441 | 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 43442 | 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 43443 | 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 43444 | 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 43445* | A frequently used special case of rspc2ev 3635 for operation values, analogous to rspceov 7203. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) ) | ||
Theorem | fnotaovb 43446 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6719. (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 43447* | An operation maps to a class to which all values belong, analogous to ffnov 7278. (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 43448 | Closure law for an operation, analogous to fovcl 7279. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶) | ||
Theorem | aovmpt4g 43449* | Value of a function given by the maps-to notation, analogous to ovmpt4g 7297. (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 43450* | Domain of closure of an operation. In contrast to oprssdm 7329, 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 43451 | The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7333 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 43452 | Reverse closure law, in contrast to ndmovrcl 7334 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 43453 | Any operation is commutative outside its domain, analogous to ndmovcom 7335. (Contributed by Alexander van der Vekens, 26-May-2017.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) ) | ||
Theorem | ndmaovass 43454 | Any operation is associative outside its domain. In contrast to ndmovass 7336 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 43455 | Any operation is distributive outside its domain. In contrast to ndmovdistr 7337 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 43368. The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6363) assures that this value is always a set, see fex 6989. 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 6700 and fvprc 6663). "(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 43367. 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 6701). To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 43457) 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 43459). 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 43472, 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 43460). 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 43508), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 43513. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 43509). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 43475. 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 6363 of (𝐹‘𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6669-> afv2eq1 43464, fveq2 6670-> afv2eq2 43465, nffv 6680-> nfafv2 43466, csbfv12 6713-> csbafv212g , rlimdm 14908-> rlimdmafv2 43506, tz6.12-1 6692-> tz6.12-1-afv2 43489, fveu 6661-> afv2eu 43486. Six theorems proved by directly using df-fv 6363 are within a mathbox (fvsb 40833, uncov 34888) or not used (rlimdmafv 43425, avril1 28242) or experimental (dfafv2 43380, dfafv22 43507). However, the remaining 11 theorems proved by directly using df-fv 6363 are used more or less often: * fvex 6683: 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 43461 resp. afv2ex 43462). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6683. * fvres 6689: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 43487). 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 43487 can be used instead of fvres 6689. * tz6.12-2 6660 (-> tz6.12-2-afv2 43485): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6663 (-> afv2prc 43474), used in 193 proofs, ** tz6.12i 6696 (-> tz6.12i-afv2 43491), used - indirectly via fvbr0 6697 and fvrn0 6698 - in 19 proofs, and in fvclss 7001 used in fvclex 7660 used in fvresex 7661 (which is not used!) and in dcomex 9869 (used in 4 proofs), ** ndmfv 6700 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6707 (-> nfunsnafv2 ), used by fvfundmfvn0 6708 (used in 3 proofs), and dffv2 6756 (not used) ** funpartfv 33406, setrec2lem1 44845 (mathboxes) * fv2 6665: only used by elfv 6668, which is only used by fv3 6688, which is not used. * dffv3 6666 (-> dfafv23 ): used by dffv4 6667 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 41282), by shftval 14433 (itself used in 11 proofs), by dffv5 33385 (mathbox) and by fvco2 6758 (-> afv2co2 43505). * fvopab5 6800: used only by ajval 28638 (not used) and by adjval 29667, which is used in adjval2 29668 (not used) and in adjbdln 29860 (used in 7 proofs). * zsum 15075: used (via isum 15076, sum0 15078, sumss 15081 and fsumsers 15085) in 76 proofs. * isumshft 15194: used in pserdv2 25018 (used in logtayl 25243, binomcxplemdvsum 40736) , eftlub 15462 (used in 4 proofs), binomcxplemnotnn0 40737 (used in binomcxp 40738 only) and logtayl 25243 (used in 4 proofs). * ovtpos 7907: used in 16 proofs. * zprod 15291: used in 3 proofs: iprod 15292, zprodn0 15293 and prodss 15301 * iprodclim3 15354: 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 6665, dffv3 6666, fvopab5 6800, zsum 15075, isumshft 15194, ovtpos 7907 and zprod 15291 are not critical or are, hopefully, also valid for the alternative definition, fvex 6683, fvres 6689 and tz6.12-2 6660 (and the theorems based on them) are essential for the current definition of function values. | ||
Syntax | cafv2 43456 | 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 43365. |
class (𝐹''''𝐴) | ||
Definition | df-afv2 43457* | 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 6363, and especially ndmfv 6700), (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 43458* | 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 43459 | 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 43460 | 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 43461 | 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 43462 | 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 43463 | Equality deduction for function value, analogous to fveq12d 6677. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵)) | ||
Theorem | afv2eq1 43464 | Equality theorem for function value, analogous to fveq1 6669. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴)) | ||
Theorem | afv2eq2 43465 | Equality theorem for function value, analogous to fveq2 6670. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵)) | ||
Theorem | nfafv2 43466 | Bound-variable hypothesis builder for function value, analogous to nffv 6680. To prove a deduction version of this analogous to nffvd 6682 is not easily possible because a deduction version of nfdfat 43375 cannot be shown easily. (Contributed by AV, 4-Sep-2022.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹''''𝐴) | ||
Theorem | csbafv212g 43467 | Move class substitution in and out of a function value, analogous to csbfv12 6713, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7198. (Contributed by AV, 4-Sep-2022.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) | ||
Theorem | fexafv2ex 43468 | 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 43469 | 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 43470 | The value of a class outside its domain is not in the range, compare with ndmfv 6700. (Contributed by AV, 2-Sep-2022.) |
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹) | ||
Theorem | funressndmafv2rn 43471 | 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 43472 | 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 43473 | If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6707. (Contributed by AV, 2-Sep-2022.) |
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹) | ||
Theorem | afv2prc 43474 | A function's value at a proper class is not defined, compare with fvprc 6663. (Contributed by AV, 5-Sep-2022.) |
⊢ (¬ 𝐴 ∈ V → (𝐹''''𝐴) ∉ ran 𝐹) | ||
Theorem | dfatafv2rnb 43475 | 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 43476 | 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 43477 | 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 43478 | 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 43479 | An alternate function value belongs to the range of the function, analogous to fvelrn 6844. (Contributed by AV, 3-Sep-2022.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹) | ||
Theorem | afv20defat 43480 | 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 43481 | An alternate function value belongs to the range of the function, analogous to fnfvelrn 6848. (Contributed by AV, 2-Sep-2022.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹) | ||
Theorem | fafv2elrn 43482 | An alternate function value belongs to the codomain of the function, analogous to ffvelrn 6849. (Contributed by AV, 2-Sep-2022.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵) | ||
Theorem | fafv2elrnb 43483 | 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 43484 | 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 43485* | Function value when 𝐹 is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6660. (Contributed by AV, 5-Sep-2022.) |
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) ∉ ran 𝐹) | ||
Theorem | afv2eu 43486* | The value of a function at a unique point, analogous to fveu 6661. (Contributed by AV, 5-Sep-2022.) |
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹''''𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) | ||
Theorem | afv2res 43487 | The value of a restricted function for an argument at which the function is defined. Analog to fvres 6689. (Contributed by AV, 5-Sep-2022.) |
⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴)) | ||
Theorem | tz6.12-afv2 43488* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6693. (Contributed by AV, 5-Sep-2022.) |
⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹''''𝐴) = 𝑦) | ||
Theorem | tz6.12-1-afv2 43489* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6692. (Contributed by AV, 5-Sep-2022.) |
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹''''𝐴) = 𝑦) | ||
Theorem | tz6.12c-afv2 43490* | Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6695. (Contributed by AV, 5-Sep-2022.) |
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) | ||
Theorem | tz6.12i-afv2 43491 | Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6696. (Contributed by AV, 5-Sep-2022.) |
⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵)) | ||
Theorem | funressnbrafv2 43492 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6716. (Contributed by AV, 7-Sep-2022.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵)) | ||
Theorem | dfatbrafv2b 43493 | Equivalence of function value and binary relation, analogous to fnbrfvb 6718 or funbrfvb 6720. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 43461). (Contributed by AV, 6-Sep-2022.) |
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
Theorem | dfatopafv2b 43494 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6719 or funopfvb 6721. (Contributed by AV, 6-Sep-2022.) |
⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
Theorem | funbrafv2 43495 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6716. (Contributed by AV, 6-Sep-2022.) |
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵)) | ||
Theorem | fnbrafv2b 43496 | Equivalence of function value and binary relation, analogous to fnbrfvb 6718. (Contributed by AV, 6-Sep-2022.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) | ||
Theorem | fnopafv2b 43497 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6719. (Contributed by AV, 6-Sep-2022.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹)) | ||
Theorem | funbrafv22b 43498 | Equivalence of function value and binary relation, analogous to funbrfvb 6720. (Contributed by AV, 6-Sep-2022.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
Theorem | funopafv2b 43499 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6721. (Contributed by AV, 6-Sep-2022.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
Theorem | dfatsnafv2 43500 | Singleton of function value, analogous to fnsnfv 6743. (Contributed by AV, 7-Sep-2022.) |
⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴})) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |