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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | aovrcl 47201 | Reverse closure for an operation value, analogous to afvvv 47157. In contrast to ovrcl 7472, 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 47202 | 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 47203 | 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 47204 | 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 47205 | 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 47206 | 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 47207 | 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 47208 | 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 47209* | A frequently used special case of rspc2ev 3635 for operation values, analogous to rspceov 7480. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑆 = ((𝑥𝐹𝑦)) ) | ||
| Theorem | fnotaovb 47210 | 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 47211* | An operation maps to a class to which all values belong, analogous to ffnov 7559. (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 47212 | Closure law for an operation, analogous to fovcl 7561. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝐶) | ||
| Theorem | aovmpt4g 47213* | Value of a function given by the maps-to notation, analogous to ovmpt4g 7580. (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 47214* | Domain of closure of an operation. In contrast to oprssdm 7614, 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 47215 | The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7618 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 47216 | Reverse closure law, in contrast to ndmovrcl 7619 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 47217 | Any operation is commutative outside its domain, analogous to ndmovcom 7620. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| ⊢ dom 𝐹 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) ) | ||
| Theorem | ndmaovass 47218 | Any operation is associative outside its domain. In contrast to ndmovass 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) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (( ((𝐴𝐹𝐵)) 𝐹𝐶)) = ((𝐴𝐹 ((𝐵𝐹𝐶)) )) ) | ||
| Theorem | ndmaovdistr 47219 | Any operation is distributive outside its domain. In contrast to ndmovdistr 7622 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 47132. The current definition of the value (𝐹‘𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 6569) assures that this value is always a set, see fex 7246. 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 47131. 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 47221) 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 47223). 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 47236, 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 47224). 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 47272), but (𝐹‘𝐴) = ∅ → (𝐹''''𝐴) ∉ ran 𝐹 is not generally valid, see afv2fv0 47277. The alternate definition, however, corresponds to the current definition ((𝐹‘𝐴) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 47273). With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 47239. 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 6569 of (𝐹‘𝐴), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6905-> afv2eq1 47228, fveq2 6906-> afv2eq2 47229, nffv 6916-> nfafv2 47230, csbfv12 6954-> csbafv212g , rlimdm 15587-> rlimdmafv2 47270, tz6.12-1 6929-> tz6.12-1-afv2 47253, fveu 6895-> afv2eu 47250. Six theorems proved by directly using df-fv 6569 are within a mathbox (fvsb 44471, uncov 37608) or not used (rlimdmafv 47189, avril1 30482) or experimental (dfafv2 47144, dfafv22 47271). However, the remaining 11 theorems proved by directly using df-fv 6569 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 47225 resp. afv2ex 47226). 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 47251). 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 47251 can be used instead of fvres 6925. * tz6.12-2 6894 (-> tz6.12-2-afv2 47249): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6898 (-> afv2prc 47238), used in 193 proofs, ** tz6.12i 6934 (-> tz6.12i-afv2 47255), used - indirectly via fvbr0 6935 and fvrn0 6936 - in 19 proofs, and in fvclss 7261 used in fvclex 7983 used in fvresex 7984 (which is not used!) and in dcomex 10487 (used in 4 proofs), ** ndmfv 6941 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6948 (-> nfunsnafv2 ), used by fvfundmfvn0 6949 (used in 3 proofs), and dffv2 7004 (not used) ** funpartfv 35946, setrec2lem1 49212 (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 44919), by shftval 15113 (itself used in 11 proofs), by dffv5 35925 (mathbox) and by fvco2 7006 (-> afv2co2 47269). * fvopab5 7049: used only by ajval 30880 (not used) and by adjval 31909, which is used in adjval2 31910 (not used) and in adjbdln 32102 (used in 7 proofs). * zsum 15754: used (via isum 15755, sum0 15757, sumss 15760 and fsumsers 15764) in 76 proofs. * isumshft 15875: used in pserdv2 26474 (used in logtayl 26702, binomcxplemdvsum 44374) , eftlub 16145 (used in 4 proofs), binomcxplemnotnn0 44375 (used in binomcxp 44376 only) and logtayl 26702 (used in 4 proofs). * ovtpos 8266: used in 16 proofs. * zprod 15973: used in 3 proofs: iprod 15974, zprodn0 15975 and prodss 15983 * iprodclim3 16036: 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 7049, zsum 15754, isumshft 15875, ovtpos 8266 and zprod 15973 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 47220 | 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 47129. |
| class (𝐹''''𝐴) | ||
| Definition | df-afv2 47221* | 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 6569, 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 47222* | 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 47223 | 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 47224 | 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 47225 | 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 47226 | 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 47227 | 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 47228 | Equality theorem for function value, analogous to fveq1 6905. (Contributed by AV, 4-Sep-2022.) |
| ⊢ (𝐹 = 𝐺 → (𝐹''''𝐴) = (𝐺''''𝐴)) | ||
| Theorem | afv2eq2 47229 | Equality theorem for function value, analogous to fveq2 6906. (Contributed by AV, 4-Sep-2022.) |
| ⊢ (𝐴 = 𝐵 → (𝐹''''𝐴) = (𝐹''''𝐵)) | ||
| Theorem | nfafv2 47230 | 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 47139 cannot be shown easily. (Contributed by AV, 4-Sep-2022.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹''''𝐴) | ||
| Theorem | csbafv212g 47231 | 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 7475. (Contributed by AV, 4-Sep-2022.) |
| ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) | ||
| Theorem | fexafv2ex 47232 | 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 47233 | 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 47234 | 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 47235 | 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 47236 | 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 47237 | 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 47238 | 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 47239 | 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 47240 | 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 47241 | 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 47242 | 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 47243 | An alternate function value belongs to the range of the function, analogous to fvelrn 7096. (Contributed by AV, 3-Sep-2022.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹) | ||
| Theorem | afv20defat 47244 | 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 47245 | An alternate function value belongs to the range of the function, analogous to fnfvelrn 7100. (Contributed by AV, 2-Sep-2022.) |
| ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ ran 𝐹) | ||
| Theorem | fafv2elcdm 47246 | An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7101. (Contributed by AV, 2-Sep-2022.) |
| ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹''''𝐶) ∈ 𝐵) | ||
| Theorem | fafv2elrnb 47247 | 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 47248 | 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 47249* | 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 47250* | 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 47251 | 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 47252* | 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 47253* | 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 47254* | 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 47255 | 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 47256 | 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 47257 | Equivalence of function value and binary relation, analogous to fnbrfvb 6959 or funbrfvb 6962. 𝐵 ∈ V is required, because otherwise 𝐴𝐹𝐵 ↔ ∅ ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐵 is always false (because of dfatafv2ex 47225). (Contributed by AV, 6-Sep-2022.) |
| ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
| Theorem | dfatopafv2b 47258 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6960 or funopfvb 6963. (Contributed by AV, 6-Sep-2022.) |
| ⊢ ((𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝐹''''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
| Theorem | funbrafv2 47259 | 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 47260 | 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 47261 | 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 47262 | Equivalence of function value and binary relation, analogous to funbrfvb 6962. (Contributed by AV, 6-Sep-2022.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | ||
| Theorem | funopafv2b 47263 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6963. (Contributed by AV, 6-Sep-2022.) |
| ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) | ||
| Theorem | dfatsnafv2 47264 | Singleton of function value, analogous to fnsnfv 6988. (Contributed by AV, 7-Sep-2022.) |
| ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴})) | ||
| Theorem | dfafv23 47265* | A definition of function value in terms of iota, analogous to dffv3 6902. (Contributed by AV, 6-Sep-2022.) |
| ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) | ||
| Theorem | dfatdmfcoafv2 47266 | Domain of a function composition, analogous to dmfco 7005. (Contributed by AV, 7-Sep-2022.) |
| ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹)) | ||
| Theorem | dfatcolem 47267* | Lemma for dfatco 47268. (Contributed by AV, 8-Sep-2022.) |
| ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ∃!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦) | ||
| Theorem | dfatco 47268 | The predicate "defined at" for a function composition. (Contributed by AV, 8-Sep-2022.) |
| ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → (𝐹 ∘ 𝐺) defAt 𝑋) | ||
| Theorem | afv2co2 47269 | Value of a function composition, analogous to fvco2 7006. (Contributed by AV, 8-Sep-2022.) |
| ⊢ ((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) → ((𝐹 ∘ 𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋))) | ||
| Theorem | rlimdmafv2 47270 | Two ways to express that a function has a limit, analogous to rlimdm 15587. (Contributed by AV, 5-Sep-2022.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 ''''𝐹))) | ||
| Theorem | dfafv22 47271 | Alternate definition of (𝐹''''𝐴) using (𝐹‘𝐴) directly. (Contributed by AV, 3-Sep-2022.) |
| ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹) | ||
| Theorem | afv2ndeffv0 47272 | If the alternate function value at an argument is undefined, i.e., not in the range of the function, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.) |
| ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹‘𝐴) = ∅) | ||
| Theorem | dfatafv2eqfv 47273 | If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.) |
| ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹‘𝐴)) | ||
| Theorem | afv2rnfveq 47274 | If the alternate function value is defined, i.e., in the range of the function, the alternate function value equals the function's value. (Contributed by AV, 3-Sep-2022.) |
| ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → (𝐹''''𝐴) = (𝐹‘𝐴)) | ||
| Theorem | afv20fv0 47275 | If the alternate function value at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022.) |
| ⊢ ((𝐹''''𝐴) = ∅ → (𝐹‘𝐴) = ∅) | ||
| Theorem | afv2fvn0fveq 47276 | If the function's value at an argument is not the empty set, it equals the alternate function value at this argument. (Contributed by AV, 3-Sep-2022.) |
| ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐹''''𝐴) = (𝐹‘𝐴)) | ||
| Theorem | afv2fv0 47277 | If the function's value at an argument is the empty set, then the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.) |
| ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) | ||
| Theorem | afv2fv0b 47278 | The function's value at an argument is the empty set if and only if the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022.) |
| ⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹)) | ||
| Theorem | afv2fv0xorb 47279 | If a set is in the range of a function, the function's value at an argument is the empty set if and only if the alternate function value at this argument is either the empty set or undefined. (Contributed by AV, 11-Sep-2022.) |
| ⊢ (∅ ∈ ran 𝐹 → ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹))) | ||
| Theorem | an4com24 47280 | Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.) |
| ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓))) | ||
| Theorem | 3an4ancom24 47281 | Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.) |
| ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓)) | ||
| Theorem | 4an21 47282 | Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.) |
| ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒 ∧ 𝜃))) | ||
| Syntax | cnelbr 47283 | Extend wff notation to include the 'not element of' relation. |
| class _∉ | ||
| Definition | df-nelbr 47284* | Define negated membership as binary relation. Analogous to df-eprel 5584 (the membership relation). (Contributed by AV, 26-Dec-2021.) |
| ⊢ _∉ = {〈𝑥, 𝑦〉 ∣ ¬ 𝑥 ∈ 𝑦} | ||
| Theorem | dfnelbr2 47285 | Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.) |
| ⊢ _∉ = ((V × V) ∖ E ) | ||
| Theorem | nelbr 47286 | The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)) | ||
| Theorem | nelbrim 47287 | If a set is related to another set by the negated membership relation, then it is not a member of the other set. The other direction of the implication is not generally true, because if 𝐴 is a proper class, then ¬ 𝐴 ∈ 𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.) |
| ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵) | ||
| Theorem | nelbrnel 47288 | A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ 𝐴 ∉ 𝐵)) | ||
| Theorem | nelbrnelim 47289 | If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.) |
| ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵) | ||
| Theorem | ralralimp 47290* | Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.) |
| ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → (∀𝑥 ∈ 𝐴 ((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏)) | ||
| Theorem | otiunsndisjX 47291* | The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
| ⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ 𝑊 {〈𝑎, 𝐵, 𝑐〉}) | ||
| Theorem | fvifeq 47292 | Equality of function values with conditional arguments, see also fvif 6922. (Contributed by Alexander van der Vekens, 21-May-2018.) |
| ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶))) | ||
| Theorem | rnfdmpr 47293 | The range of a one-to-one function 𝐹 of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝐹 Fn {𝑋, 𝑌} → ran 𝐹 = {(𝐹‘𝑋), (𝐹‘𝑌)})) | ||
| Theorem | imarnf1pr 47294 | The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function from a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.) |
| ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (((𝐹:{𝑋, 𝑌}⟶dom 𝐸 ∧ 𝐸:dom 𝐸⟶𝑅) ∧ ((𝐸‘(𝐹‘𝑋)) = 𝐴 ∧ (𝐸‘(𝐹‘𝑌)) = 𝐵)) → (𝐸 “ ran 𝐹) = {𝐴, 𝐵})) | ||
| Theorem | funop1 47295* | A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Avoid depending on this detail.) |
| ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) | ||
| Theorem | fun2dmnopgexmpl 47296 | A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Avoid depending on this detail.) |
| ⊢ (𝐺 = {〈0, 1〉, 〈1, 1〉} → ¬ 𝐺 ∈ (V × V)) | ||
| Theorem | opabresex0d 47297* | A collection of ordered pairs, the class of all possible second components being a set, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 1-Jan-2021.) |
| ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) | ||
| Theorem | opabbrfex0d 47298* | A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.) |
| ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝜃) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → {𝑦 ∣ 𝜃} ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ∈ V) | ||
| Theorem | opabresexd 47299* | A collection of ordered pairs, the second component being a function, with a restriction of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) |
| ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} ∈ V) | ||
| Theorem | opabbrfexd 47300* | A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.) |
| ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥𝑅𝑦) → 𝑦:𝐴⟶𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ∈ V) | ||
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