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Statement | ||
In the following, a second approach is followed to define function values alternately to df-afv 45913. The current definition of the value (πΉβπ΄) of a function πΉ at an argument π΄ (see df-fv 6551) assures that this value is always a set, see fex 7230. This is because this definition can be applied to any classes πΉ and π΄, and evaluates to the empty set when it is not meaningful (as shown by ndmfv 6926 and fvprc 6883). "(πΉβπ΄) is meaningful" means "the class πΉ regarded as function is defined at the argument π΄" in this context. This is also expressed by πΉ defAt π΄, see df-dfat 45912. In the theory of partial functions, it is a common case that πΉ is not defined at π΄. Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (πΉβπ΄) = β alone it cannot be decided/derived whether (πΉβπ΄) is meaningful (πΉ is actually a function which is defined for π΄ and really has the function value β at π΄) or not. Therefore, additional assumptions are required, such as β β ran πΉ, β β ran πΉ, πΉ defAt π΄, or Fun πΉ β§ π΄ β dom πΉ (see, for example, ndmfvrcl 6927). To avoid such an ambiguity, an alternative definition (πΉ''''π΄) (see df-afv2 46002) would be possible which evaluates to a set not belonging to the range of πΉ ((πΉ''''π΄) = π« βͺ ran πΉ) if it is not meaningful (see ndfatafv2 46004). We say "(πΉ''''π΄) is not defined (or undefined)" if (πΉ''''π΄) is not in the range of πΉ ((πΉ''''π΄) β ran πΉ). Because of afv2ndefb 46017, this is equivalent to ((πΉ''''π΄) = π« βͺ ran πΉ. If (πΉ''''π΄) is in the range of πΉ ((πΉ''''π΄) β ran πΉ), we say that "(πΉ''''π΄) is defined". If ran πΉ is a set, we can use the symbol Undef to express that (πΉ''''π΄) is not defined: (πΉ''''π΄) = (Undefβran πΉ) (see ndfatafv2undef 46005). We could have used this symbol directly to define the alternate value of a function, which would have the advantage that (πΉ''''π΄) 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 πΉ β V in most of the theorems. To summarize, that means (πΉ''''π΄) β ran πΉ β (πΉβπ΄) = β (see afv2ndeffv0 46053), but (πΉβπ΄) = β β (πΉ''''π΄) β ran πΉ is not generally valid, see afv2fv0 46058. The alternate definition, however, corresponds to the current definition ((πΉβπ΄) = (πΉ''''π΄)) if the function πΉ is defined at π΄ (see dfatafv2eqfv 46054). With this definition the following intuitive equivalence holds: (πΉ defAt π΄ β (πΉ''''π΄) β ran πΉ), see dfatafv2rnb 46020. An interesting question would be if (πΉβπ΄) could be replaced by (πΉ'''π΄) in most of the theorems based on function values. If we look at the (currently 24) proofs using the definition df-fv 6551 of (πΉβπ΄), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6890-> afv2eq1 46009, fveq2 6891-> afv2eq2 46010, nffv 6901-> nfafv2 46011, csbfv12 6939-> csbafv212g , rlimdm 15497-> rlimdmafv2 46051, tz6.12-1 6914-> tz6.12-1-afv2 46034, fveu 6880-> afv2eu 46031. Six theorems proved by directly using df-fv 6551 are within a mathbox (fvsb 43299, uncov 36561) or not used (rlimdmafv 45970, avril1 29754) or experimental (dfafv2 45925, dfafv22 46052). However, the remaining 11 theorems proved by directly using df-fv 6551 are used more or less often: * fvex 6904: 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 46006 resp. afv2ex 46007). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6904. * fvres 6910: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 46032). In the undefined case such a theorem cannot exist (without additional assumtions), because the range of (πΉ βΎ π΅) is mostly different from the range of πΉ, and therefore also the "undefined" values are different. All of these 400 proofs have to be checked if afv2res 46032 can be used instead of fvres 6910. * tz6.12-2 6879 (-> tz6.12-2-afv2 46030): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6883 (-> afv2prc 46019), used in 193 proofs, ** tz6.12i 6919 (-> tz6.12i-afv2 46036), used - indirectly via fvbr0 6920 and fvrn0 6921 - in 19 proofs, and in fvclss 7243 used in fvclex 7947 used in fvresex 7948 (which is not used!) and in dcomex 10444 (used in 4 proofs), ** ndmfv 6926 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6933 (-> nfunsnafv2 ), used by fvfundmfvn0 6934 (used in 3 proofs), and dffv2 6986 (not used) ** funpartfv 34992, setrec2lem1 47822 (mathboxes) * fv2 6886: only used by elfv 6889, which is only used by fv3 6909, which is not used. * dffv3 6887 (-> dfafv23 ): used by dffv4 6888 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 43748), by shftval 15023 (itself used in 11 proofs), by dffv5 34971 (mathbox) and by fvco2 6988 (-> afv2co2 46050). * fvopab5 7030: used only by ajval 30152 (not used) and by adjval 31181, which is used in adjval2 31182 (not used) and in adjbdln 31374 (used in 7 proofs). * zsum 15666: used (via isum 15667, sum0 15669, sumss 15672 and fsumsers 15676) in 76 proofs. * isumshft 15787: used in pserdv2 25949 (used in logtayl 26175, binomcxplemdvsum 43202) , eftlub 16054 (used in 4 proofs), binomcxplemnotnn0 43203 (used in binomcxp 43204 only) and logtayl 26175 (used in 4 proofs). * ovtpos 8228: used in 16 proofs. * zprod 15883: used in 3 proofs: iprod 15884, zprodn0 15885 and prodss 15893 * iprodclim3 15946: 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 6886, dffv3 6887, fvopab5 7030, zsum 15666, isumshft 15787, ovtpos 8228 and zprod 15883 are not critical or are, hopefully, also valid for the alternative definition, fvex 6904, fvres 6910 and tz6.12-2 6879 (and the theorems based on them) are essential for the current definition of function values. | ||
Syntax | cafv2 46001 | Extend the definition of a class to include the alternate function value. Read: "the value of πΉ at π΄ " or "πΉ of π΄". For using several apostrophes as a symbol see comment for cafv 45910. |
class (πΉ''''π΄) | ||
Definition | df-afv2 46002* | Alternate definition of the value of a function, (πΉ''''π΄), also known as function application (and called "alternate function value" in the following). In contrast to (πΉβπ΄) = β (see comment of df-fv 6551, and especially ndmfv 6926), (πΉ''''π΄) is guaranteed not to be in the range of πΉ if πΉ is not defined at π΄ (whereas β can be a member of ran πΉ). (Contributed by AV, 2-Sep-2022.) |
β’ (πΉ''''π΄) = if(πΉ defAt π΄, (β©π₯π΄πΉπ₯), π« βͺ ran πΉ) | ||
Theorem | dfatafv2iota 46003* | If a function is defined at a class π΄ the alternate function value at π΄ is the unique value assigned to π΄ by the function (analogously to (πΉβπ΄)). (Contributed by AV, 2-Sep-2022.) |
β’ (πΉ defAt π΄ β (πΉ''''π΄) = (β©π₯π΄πΉπ₯)) | ||
Theorem | ndfatafv2 46004 | The alternate function value at a class π΄ if the function is not defined at this set π΄. (Contributed by AV, 2-Sep-2022.) |
β’ (Β¬ πΉ defAt π΄ β (πΉ''''π΄) = π« βͺ ran πΉ) | ||
Theorem | ndfatafv2undef 46005 | The alternate function value at a class π΄ is undefined if the function, whose range is a set, is not defined at π΄. (Contributed by AV, 2-Sep-2022.) |
β’ ((ran πΉ β π β§ Β¬ πΉ defAt π΄) β (πΉ''''π΄) = (Undefβran πΉ)) | ||
Theorem | dfatafv2ex 46006 | The alternate function value at a class π΄ is always a set if the function/class πΉ is defined at π΄. (Contributed by AV, 6-Sep-2022.) |
β’ (πΉ defAt π΄ β (πΉ''''π΄) β V) | ||
Theorem | afv2ex 46007 | The alternate function value is always a set if the range of the function is a set. (Contributed by AV, 2-Sep-2022.) |
β’ (ran πΉ β π β (πΉ''''π΄) β V) | ||
Theorem | afv2eq12d 46008 | Equality deduction for function value, analogous to fveq12d 6898. (Contributed by AV, 4-Sep-2022.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) β β’ (π β (πΉ''''π΄) = (πΊ''''π΅)) | ||
Theorem | afv2eq1 46009 | Equality theorem for function value, analogous to fveq1 6890. (Contributed by AV, 4-Sep-2022.) |
β’ (πΉ = πΊ β (πΉ''''π΄) = (πΊ''''π΄)) | ||
Theorem | afv2eq2 46010 | Equality theorem for function value, analogous to fveq2 6891. (Contributed by AV, 4-Sep-2022.) |
β’ (π΄ = π΅ β (πΉ''''π΄) = (πΉ''''π΅)) | ||
Theorem | nfafv2 46011 | Bound-variable hypothesis builder for function value, analogous to nffv 6901. To prove a deduction version of this analogous to nffvd 6903 is not easily possible because a deduction version of nfdfat 45920 cannot be shown easily. (Contributed by AV, 4-Sep-2022.) |
β’ β²π₯πΉ & β’ β²π₯π΄ β β’ β²π₯(πΉ''''π΄) | ||
Theorem | csbafv212g 46012 | Move class substitution in and out of a function value, analogous to csbfv12 6939, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7453. (Contributed by AV, 4-Sep-2022.) |
β’ (π΄ β π β β¦π΄ / π₯β¦(πΉ''''π΅) = (β¦π΄ / π₯β¦πΉ''''β¦π΄ / π₯β¦π΅)) | ||
Theorem | fexafv2ex 46013 | 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.) |
β’ (πΉ β π β (πΉ''''π΄) β V) | ||
Theorem | ndfatafv2nrn 46014 | The alternate function value at a class π΄ at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022.) |
β’ (Β¬ πΉ defAt π΄ β (πΉ''''π΄) β ran πΉ) | ||
Theorem | ndmafv2nrn 46015 | The value of a class outside its domain is not in the range, compare with ndmfv 6926. (Contributed by AV, 2-Sep-2022.) |
β’ (Β¬ π΄ β dom πΉ β (πΉ''''π΄) β ran πΉ) | ||
Theorem | funressndmafv2rn 46016 | The alternate function value at a class π΄ is defined, i.e., in the range of the function if the function is defined at π΄. (Contributed by AV, 2-Sep-2022.) |
β’ (πΉ defAt π΄ β (πΉ''''π΄) β ran πΉ) | ||
Theorem | afv2ndefb 46017 | Two ways to say that an alternate function value is not defined. (Contributed by AV, 5-Sep-2022.) |
β’ ((πΉ''''π΄) = π« βͺ ran πΉ β (πΉ''''π΄) β ran πΉ) | ||
Theorem | nfunsnafv2 46018 | If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6933. (Contributed by AV, 2-Sep-2022.) |
β’ (Β¬ Fun (πΉ βΎ {π΄}) β (πΉ''''π΄) β ran πΉ) | ||
Theorem | afv2prc 46019 | A function's value at a proper class is not defined, compare with fvprc 6883. (Contributed by AV, 5-Sep-2022.) |
β’ (Β¬ π΄ β V β (πΉ''''π΄) β ran πΉ) | ||
Theorem | dfatafv2rnb 46020 | The alternate function value at a class π΄ is defined, i.e. in the range of the function, iff the function is defined at π΄. (Contributed by AV, 2-Sep-2022.) |
β’ (πΉ defAt π΄ β (πΉ''''π΄) β ran πΉ) | ||
Theorem | afv2orxorb 46021 | If a set is in the range of a function, the alternate function value at a class π΄ equals this set or is not in the range of the function iff the alternate function value at the class π΄ either equals this set or is not in the range of the function. If π΅ β ran πΉ, both disjuncts of the exclusive or can be true: (πΉ''''π΄) = π΅ β (πΉ''''π΄) β ran πΉ. (Contributed by AV, 11-Sep-2022.) |
β’ (π΅ β ran πΉ β (((πΉ''''π΄) = π΅ β¨ (πΉ''''π΄) β ran πΉ) β ((πΉ''''π΄) = π΅ β» (πΉ''''π΄) β ran πΉ))) | ||
Theorem | dmafv2rnb 46022 | The alternate function value at a class π΄ is defined, i.e., in the range of the function, iff π΄ is in the domain of the function. (Contributed by AV, 3-Sep-2022.) |
β’ (Fun (πΉ βΎ {π΄}) β (π΄ β dom πΉ β (πΉ''''π΄) β ran πΉ)) | ||
Theorem | fundmafv2rnb 46023 | The alternate function value at a class π΄ is defined, i.e., in the range of the function iff π΄ is in the domain of the function. (Contributed by AV, 3-Sep-2022.) |
β’ (Fun πΉ β (π΄ β dom πΉ β (πΉ''''π΄) β ran πΉ)) | ||
Theorem | afv2elrn 46024 | An alternate function value belongs to the range of the function, analogous to fvelrn 7078. (Contributed by AV, 3-Sep-2022.) |
β’ ((Fun πΉ β§ π΄ β dom πΉ) β (πΉ''''π΄) β ran πΉ) | ||
Theorem | afv20defat 46025 | 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.) |
β’ ((πΉ''''π΄) = β β πΉ defAt π΄) | ||
Theorem | fnafv2elrn 46026 | An alternate function value belongs to the range of the function, analogous to fnfvelrn 7082. (Contributed by AV, 2-Sep-2022.) |
β’ ((πΉ Fn π΄ β§ π΅ β π΄) β (πΉ''''π΅) β ran πΉ) | ||
Theorem | fafv2elcdm 46027 | An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7083. (Contributed by AV, 2-Sep-2022.) |
β’ ((πΉ:π΄βΆπ΅ β§ πΆ β π΄) β (πΉ''''πΆ) β π΅) | ||
Theorem | fafv2elrnb 46028 | 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.) |
β’ (πΉ:π΄βΆπ΅ β (πΆ β π΄ β (πΉ''''πΆ) β ran πΉ)) | ||
Theorem | fcdmvafv2v 46029 | If the codomain of a function is a set, the alternate function value is always also a set. (Contributed by AV, 4-Sep-2022.) |
β’ ((πΉ:π΄βΆπ΅ β§ π΅ β π) β (πΉ''''πΆ) β V) | ||
Theorem | tz6.12-2-afv2 46030* | Function value when πΉ is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6879. (Contributed by AV, 5-Sep-2022.) |
β’ (Β¬ β!π₯ π΄πΉπ₯ β (πΉ''''π΄) β ran πΉ) | ||
Theorem | afv2eu 46031* | The value of a function at a unique point, analogous to fveu 6880. (Contributed by AV, 5-Sep-2022.) |
β’ (β!π₯ π΄πΉπ₯ β (πΉ''''π΄) = βͺ {π₯ β£ π΄πΉπ₯}) | ||
Theorem | afv2res 46032 | The value of a restricted function for an argument at which the function is defined. Analog to fvres 6910. (Contributed by AV, 5-Sep-2022.) |
β’ ((πΉ defAt π΄ β§ π΄ β π΅) β ((πΉ βΎ π΅)''''π΄) = (πΉ''''π΄)) | ||
Theorem | tz6.12-afv2 46033* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6916. (Contributed by AV, 5-Sep-2022.) |
β’ ((β¨π΄, π¦β© β πΉ β§ β!π¦β¨π΄, π¦β© β πΉ) β (πΉ''''π΄) = π¦) | ||
Theorem | tz6.12-1-afv2 46034* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6914. (Contributed by AV, 5-Sep-2022.) |
β’ ((π΄πΉπ¦ β§ β!π¦ π΄πΉπ¦) β (πΉ''''π΄) = π¦) | ||
Theorem | tz6.12c-afv2 46035* | Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6913. (Contributed by AV, 5-Sep-2022.) |
β’ (β!π¦ π΄πΉπ¦ β ((πΉ''''π΄) = π¦ β π΄πΉπ¦)) | ||
Theorem | tz6.12i-afv2 46036 | Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6919. (Contributed by AV, 5-Sep-2022.) |
β’ (π΅ β ran πΉ β ((πΉ''''π΄) = π΅ β π΄πΉπ΅)) | ||
Theorem | funressnbrafv2 46037 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6942. (Contributed by AV, 7-Sep-2022.) |
β’ (((π΄ β π β§ π΅ β π) β§ Fun (πΉ βΎ {π΄})) β (π΄πΉπ΅ β (πΉ''''π΄) = π΅)) | ||
Theorem | dfatbrafv2b 46038 | Equivalence of function value and binary relation, analogous to fnbrfvb 6944 or funbrfvb 6946. π΅ β V is required, because otherwise π΄πΉπ΅ β β β πΉ can be true, but (πΉ''''π΄) = π΅ is always false (because of dfatafv2ex 46006). (Contributed by AV, 6-Sep-2022.) |
β’ ((πΉ defAt π΄ β§ π΅ β π) β ((πΉ''''π΄) = π΅ β π΄πΉπ΅)) | ||
Theorem | dfatopafv2b 46039 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6945 or funopfvb 6947. (Contributed by AV, 6-Sep-2022.) |
β’ ((πΉ defAt π΄ β§ π΅ β π) β ((πΉ''''π΄) = π΅ β β¨π΄, π΅β© β πΉ)) | ||
Theorem | funbrafv2 46040 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6942. (Contributed by AV, 6-Sep-2022.) |
β’ (Fun πΉ β (π΄πΉπ΅ β (πΉ''''π΄) = π΅)) | ||
Theorem | fnbrafv2b 46041 | Equivalence of function value and binary relation, analogous to fnbrfvb 6944. (Contributed by AV, 6-Sep-2022.) |
β’ ((πΉ Fn π΄ β§ π΅ β π΄) β ((πΉ''''π΅) = πΆ β π΅πΉπΆ)) | ||
Theorem | fnopafv2b 46042 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6945. (Contributed by AV, 6-Sep-2022.) |
β’ ((πΉ Fn π΄ β§ π΅ β π΄) β ((πΉ''''π΅) = πΆ β β¨π΅, πΆβ© β πΉ)) | ||
Theorem | funbrafv22b 46043 | Equivalence of function value and binary relation, analogous to funbrfvb 6946. (Contributed by AV, 6-Sep-2022.) |
β’ ((Fun πΉ β§ π΄ β dom πΉ) β ((πΉ''''π΄) = π΅ β π΄πΉπ΅)) | ||
Theorem | funopafv2b 46044 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6947. (Contributed by AV, 6-Sep-2022.) |
β’ ((Fun πΉ β§ π΄ β dom πΉ) β ((πΉ''''π΄) = π΅ β β¨π΄, π΅β© β πΉ)) | ||
Theorem | dfatsnafv2 46045 | Singleton of function value, analogous to fnsnfv 6970. (Contributed by AV, 7-Sep-2022.) |
β’ (πΉ defAt π΄ β {(πΉ''''π΄)} = (πΉ β {π΄})) | ||
Theorem | dfafv23 46046* | A definition of function value in terms of iota, analogous to dffv3 6887. (Contributed by AV, 6-Sep-2022.) |
β’ (πΉ defAt π΄ β (πΉ''''π΄) = (β©π₯π₯ β (πΉ β {π΄}))) | ||
Theorem | dfatdmfcoafv2 46047 | Domain of a function composition, analogous to dmfco 6987. (Contributed by AV, 7-Sep-2022.) |
β’ (πΊ defAt π΄ β (π΄ β dom (πΉ β πΊ) β (πΊ''''π΄) β dom πΉ)) | ||
Theorem | dfatcolem 46048* | Lemma for dfatco 46049. (Contributed by AV, 8-Sep-2022.) |
β’ ((πΊ defAt π β§ πΉ defAt (πΊ''''π)) β β!π¦ π(πΉ β πΊ)π¦) | ||
Theorem | dfatco 46049 | The predicate "defined at" for a function composition. (Contributed by AV, 8-Sep-2022.) |
β’ ((πΊ defAt π β§ πΉ defAt (πΊ''''π)) β (πΉ β πΊ) defAt π) | ||
Theorem | afv2co2 46050 | Value of a function composition, analogous to fvco2 6988. (Contributed by AV, 8-Sep-2022.) |
β’ ((πΊ defAt π β§ πΉ defAt (πΊ''''π)) β ((πΉ β πΊ)''''π) = (πΉ''''(πΊ''''π))) | ||
Theorem | rlimdmafv2 46051 | Two ways to express that a function has a limit, analogous to rlimdm 15497. (Contributed by AV, 5-Sep-2022.) |
β’ (π β πΉ:π΄βΆβ) & β’ (π β sup(π΄, β*, < ) = +β) β β’ (π β (πΉ β dom βπ β πΉ βπ ( βπ ''''πΉ))) | ||
Theorem | dfafv22 46052 | Alternate definition of (πΉ''''π΄) using (πΉβπ΄) directly. (Contributed by AV, 3-Sep-2022.) |
β’ (πΉ''''π΄) = if(πΉ defAt π΄, (πΉβπ΄), π« βͺ ran πΉ) | ||
Theorem | afv2ndeffv0 46053 | 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.) |
β’ ((πΉ''''π΄) β ran πΉ β (πΉβπ΄) = β ) | ||
Theorem | dfatafv2eqfv 46054 | If a function is defined at a class π΄, the alternate function value equals the function's value at π΄. (Contributed by AV, 3-Sep-2022.) |
β’ (πΉ defAt π΄ β (πΉ''''π΄) = (πΉβπ΄)) | ||
Theorem | afv2rnfveq 46055 | 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.) |
β’ ((πΉ''''π΄) β ran πΉ β (πΉ''''π΄) = (πΉβπ΄)) | ||
Theorem | afv20fv0 46056 | 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.) |
β’ ((πΉ''''π΄) = β β (πΉβπ΄) = β ) | ||
Theorem | afv2fvn0fveq 46057 | 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.) |
β’ ((πΉβπ΄) β β β (πΉ''''π΄) = (πΉβπ΄)) | ||
Theorem | afv2fv0 46058 | 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.) |
β’ ((πΉβπ΄) = β β ((πΉ''''π΄) = β β¨ (πΉ''''π΄) β ran πΉ)) | ||
Theorem | afv2fv0b 46059 | 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.) |
β’ ((πΉβπ΄) = β β ((πΉ''''π΄) = β β¨ (πΉ''''π΄) β ran πΉ)) | ||
Theorem | afv2fv0xorb 46060 | 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.) |
β’ (β β ran πΉ β ((πΉβπ΄) = β β ((πΉ''''π΄) = β β» (πΉ''''π΄) β ran πΉ))) | ||
Theorem | an4com24 46061 | Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.) |
β’ (((π β§ π) β§ (π β§ π)) β ((π β§ π) β§ (π β§ π))) | ||
Theorem | 3an4ancom24 46062 | Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.) |
β’ (((π β§ π β§ π) β§ π) β ((π β§ π β§ π) β§ π)) | ||
Theorem | 4an21 46063 | Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.) |
β’ (((π β§ π) β§ π β§ π) β (π β§ (π β§ π β§ π))) | ||
Syntax | cnelbr 46064 | Extend wff notation to include the 'not elemet of' relation. |
class _β | ||
Definition | df-nelbr 46065* | Define negated membership as binary relation. Analogous to df-eprel 5580 (the membership relation). (Contributed by AV, 26-Dec-2021.) |
β’ _β = {β¨π₯, π¦β© β£ Β¬ π₯ β π¦} | ||
Theorem | dfnelbr2 46066 | Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.) |
β’ _β = ((V Γ V) β E ) | ||
Theorem | nelbr 46067 | The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.) |
β’ ((π΄ β π β§ π΅ β π) β (π΄ _β π΅ β Β¬ π΄ β π΅)) | ||
Theorem | nelbrim 46068 | 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 π΄ is a proper class, then Β¬ π΄ β π΅ would be true, but not π΄ _β π΅. (Contributed by AV, 26-Dec-2021.) |
β’ (π΄ _β π΅ β Β¬ π΄ β π΅) | ||
Theorem | nelbrnel 46069 | 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.) |
β’ ((π΄ β π β§ π΅ β π) β (π΄ _β π΅ β π΄ β π΅)) | ||
Theorem | nelbrnelim 46070 | 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.) |
β’ (π΄ _β π΅ β π΄ β π΅) | ||
Theorem | ralralimp 46071* | 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.) |
β’ ((π β§ π΄ β β ) β (βπ₯ β π΄ ((π β (π β¨ π)) β§ Β¬ π) β π)) | ||
Theorem | otiunsndisjX 46072* | 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.) |
β’ (π΅ β π β Disj π β π βͺ π β π {β¨π, π΅, πβ©}) | ||
Theorem | fvifeq 46073 | Equality of function values with conditional arguments, see also fvif 6907. (Contributed by Alexander van der Vekens, 21-May-2018.) |
β’ (π΄ = if(π, π΅, πΆ) β (πΉβπ΄) = if(π, (πΉβπ΅), (πΉβπΆ))) | ||
Theorem | rnfdmpr 46074 | The range of a one-to-one function πΉ of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.) |
β’ ((π β π β§ π β π) β (πΉ Fn {π, π} β ran πΉ = {(πΉβπ), (πΉβπ)})) | ||
Theorem | imarnf1pr 46075 | The image of the range of a function πΉ under a function πΈ if πΉ is a function from a pair into the domain of πΈ. (Contributed by Alexander van der Vekens, 2-Feb-2018.) |
β’ ((π β π β§ π β π) β (((πΉ:{π, π}βΆdom πΈ β§ πΈ:dom πΈβΆπ ) β§ ((πΈβ(πΉβπ)) = π΄ β§ (πΈβ(πΉβπ)) = π΅)) β (πΈ β ran πΉ) = {π΄, π΅})) | ||
Theorem | funop1 46076* | 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.) |
β’ (βπ₯βπ¦ πΉ = β¨π₯, π¦β© β (Fun πΉ β βπ₯βπ¦ πΉ = {β¨π₯, π¦β©})) | ||
Theorem | fun2dmnopgexmpl 46077 | 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.) |
β’ (πΊ = {β¨0, 1β©, β¨1, 1β©} β Β¬ πΊ β (V Γ V)) | ||
Theorem | opabresex0d 46078* | 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.) |
β’ ((π β§ π₯π π¦) β π₯ β πΆ) & β’ ((π β§ π₯π π¦) β π) & β’ ((π β§ π₯ β πΆ) β {π¦ β£ π} β π) & β’ (π β πΆ β π) β β’ (π β {β¨π₯, π¦β© β£ (π₯π π¦ β§ π)} β V) | ||
Theorem | opabbrfex0d 46079* | A collection of ordered pairs, the class of all possible second components being a set, is a set. (Contributed by AV, 15-Jan-2021.) |
β’ ((π β§ π₯π π¦) β π₯ β πΆ) & β’ ((π β§ π₯π π¦) β π) & β’ ((π β§ π₯ β πΆ) β {π¦ β£ π} β π) & β’ (π β πΆ β π) β β’ (π β {β¨π₯, π¦β© β£ π₯π π¦} β V) | ||
Theorem | opabresexd 46080* | 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.) |
β’ ((π β§ π₯π π¦) β π₯ β πΆ) & β’ ((π β§ π₯π π¦) β π¦:π΄βΆπ΅) & β’ ((π β§ π₯ β πΆ) β π΄ β π) & β’ ((π β§ π₯ β πΆ) β π΅ β π) & β’ (π β πΆ β π) β β’ (π β {β¨π₯, π¦β© β£ (π₯π π¦ β§ π)} β V) | ||
Theorem | opabbrfexd 46081* | A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.) |
β’ ((π β§ π₯π π¦) β π₯ β πΆ) & β’ ((π β§ π₯π π¦) β π¦:π΄βΆπ΅) & β’ ((π β§ π₯ β πΆ) β π΄ β π) & β’ ((π β§ π₯ β πΆ) β π΅ β π) & β’ (π β πΆ β π) β β’ (π β {β¨π₯, π¦β© β£ π₯π π¦} β V) | ||
Theorem | f1oresf1orab 46082* | Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.) |
β’ πΉ = (π₯ β π΄ β¦ πΆ) & β’ (π β πΉ:π΄β1-1-ontoβπ΅) & β’ (π β π· β π΄) & β’ ((π β§ π₯ β π΄ β§ π¦ = πΆ) β (π β π₯ β π·)) β β’ (π β (πΉ βΎ π·):π·β1-1-ontoβ{π¦ β π΅ β£ π}) | ||
Theorem | f1oresf1o 46083* | Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.) |
β’ (π β πΉ:π΄β1-1-ontoβπ΅) & β’ (π β π· β π΄) & β’ (π β (βπ₯ β π· (πΉβπ₯) = π¦ β (π¦ β π΅ β§ π))) β β’ (π β (πΉ βΎ π·):π·β1-1-ontoβ{π¦ β π΅ β£ π}) | ||
Theorem | f1oresf1o2 46084* | Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.) |
β’ (π β πΉ:π΄β1-1-ontoβπ΅) & β’ (π β π· β π΄) & β’ ((π β§ π¦ = (πΉβπ₯)) β (π₯ β π· β π)) β β’ (π β (πΉ βΎ π·):π·β1-1-ontoβ{π¦ β π΅ β£ π}) | ||
Theorem | fvmptrab 46085* | Value of a function mapping a set to a class abstraction restricting a class depending on the argument of the function. More general version of fvmptrabfv 7029, but relying on the fact that out-of-domain arguments evaluate to the empty set, which relies on set.mm's particular encoding. (Contributed by AV, 14-Feb-2022.) |
β’ πΉ = (π₯ β π β¦ {π¦ β π β£ π}) & β’ (π₯ = π β (π β π)) & β’ (π₯ = π β π = π) & β’ (π β π β π β V) & β’ (π β π β π = β ) β β’ (πΉβπ) = {π¦ β π β£ π} | ||
Theorem | fvmptrabdm 46086* | Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 7029. (Suggested by BJ, 18-Feb-2022.) (Contributed by AV, 18-Feb-2022.) |
β’ πΉ = (π₯ β π β¦ {π¦ β (πΊβπ) β£ π}) & β’ (π₯ = π β (π β π)) & β’ (π β dom πΊ β π β dom πΉ) β β’ (πΉβπ) = {π¦ β (πΊβπ) β£ π} | ||
Theorem | cnambpcma 46087 | ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (((π΄ β π΅) + πΆ) β π΄) = (πΆ β π΅)) | ||
Theorem | cnapbmcpd 46088 | ((a+b)-c)+d = ((a+d)+b)-c holds for complex numbers a,b,c,d. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β (((π΄ + π΅) β πΆ) + π·) = (((π΄ + π·) + π΅) β πΆ)) | ||
Theorem | addsubeq0 46089 | The sum of two complex numbers is equal to the difference of these two complex numbers iff the subtrahend is 0. (Contributed by AV, 8-May-2023.) |
β’ ((π΄ β β β§ π΅ β β) β ((π΄ + π΅) = (π΄ β π΅) β π΅ = 0)) | ||
Theorem | leaddsuble 46090 | Addition and subtraction on one side of "less than or equal to". (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (π΅ β€ πΆ β ((π΄ + π΅) β πΆ) β€ π΄)) | ||
Theorem | 2leaddle2 46091 | If two real numbers are less than a third real number, the sum of the real numbers is less than twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ < πΆ β§ π΅ < πΆ) β (π΄ + π΅) < (2 Β· πΆ))) | ||
Theorem | ltnltne 46092 | Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ < π΅ β (Β¬ π΅ < π΄ β§ Β¬ π΅ = π΄))) | ||
Theorem | p1lep2 46093 | A real number increasd by 1 is less than or equal to the number increased by 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
β’ (π β β β (π + 1) β€ (π + 2)) | ||
Theorem | ltsubsubaddltsub 46094 | If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.) |
β’ ((π½ β β β§ (πΏ β β β§ π β β β§ π β β)) β (π½ < ((πΏ β π) β π) β (π½ + π) < (πΏ β π))) | ||
Theorem | zm1nn 46095 | An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.) |
β’ ((π β β0 β§ πΏ β β€) β ((π½ β β β§ 0 β€ π½ β§ π½ < ((πΏ β π) β 1)) β (πΏ β 1) β β)) | ||
Theorem | readdcnnred 46096 | The sum of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β (β β β)) β β’ (π β (π΄ + π΅) β β) | ||
Theorem | resubcnnred 46097 | The difference of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β (β β β)) β β’ (π β (π΄ β π΅) β β) | ||
Theorem | recnmulnred 46098 | The product of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β (β β β)) & β’ (π β π΄ β 0) β β’ (π β (π΄ Β· π΅) β β) | ||
Theorem | cndivrenred 46099 | The quotient of an imaginary number and a real number is not a real number. (Contributed by AV, 23-Jan-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β (β β β)) & β’ (π β π΄ β 0) β β’ (π β (π΅ / π΄) β β) | ||
Theorem | sqrtnegnre 46100 | The square root of a negative number is not a real number. (Contributed by AV, 28-Feb-2023.) |
β’ ((π β β β§ π < 0) β (ββπ) β β) |
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