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Type | Label | Description |
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Statement | ||
Theorem | aovovn0oveq 46201 | 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.) |
β’ ((π΄πΉπ΅) β β β ((π΄πΉπ΅)) = (π΄πΉπ΅)) | ||
Theorem | aov0nbovbi 46202 | 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.) |
β’ (β β πΆ β ( ((π΄πΉπ΅)) β πΆ β (π΄πΉπ΅) β πΆ)) | ||
Theorem | aovov0bi 46203 | 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.) |
β’ ((π΄πΉπ΅) = β β ( ((π΄πΉπ΅)) = β β¨ ((π΄πΉπ΅)) = V)) | ||
Theorem | rspceaov 46204* | A frequently used special case of rspc2ev 3624 for operation values, analogous to rspceov 7459. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ ((πΆ β π΄ β§ π· β π΅ β§ π = ((πΆπΉπ·)) ) β βπ₯ β π΄ βπ¦ β π΅ π = ((π₯πΉπ¦)) ) | ||
Theorem | fnotaovb 46205 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6945. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ ((πΉ Fn (π΄ Γ π΅) β§ πΆ β π΄ β§ π· β π΅) β ( ((πΆπΉπ·)) = π β β¨πΆ, π·, π β© β πΉ)) | ||
Theorem | ffnaov 46206* | An operation maps to a class to which all values belong, analogous to ffnov 7538. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ (πΉ:(π΄ Γ π΅)βΆπΆ β (πΉ Fn (π΄ Γ π΅) β§ βπ₯ β π΄ βπ¦ β π΅ ((π₯πΉπ¦)) β πΆ)) | ||
Theorem | faovcl 46207 | Closure law for an operation, analogous to fovcl 7540. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ πΉ:(π Γ π)βΆπΆ β β’ ((π΄ β π β§ π΅ β π) β ((π΄πΉπ΅)) β πΆ) | ||
Theorem | aovmpt4g 46208* | Value of a function given by the maps-to notation, analogous to ovmpt4g 7558. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ πΉ = (π₯ β π΄, π¦ β π΅ β¦ πΆ) β β’ ((π₯ β π΄ β§ π¦ β π΅ β§ πΆ β π) β ((π₯πΉπ¦)) = πΆ) | ||
Theorem | aoprssdm 46209* | Domain of closure of an operation. In contrast to oprssdm 7592, no additional property for S (Β¬ β β π) is required! (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ ((π₯ β π β§ π¦ β π) β ((π₯πΉπ¦)) β π) β β’ (π Γ π) β dom πΉ | ||
Theorem | ndmaovcl 46210 | The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7596 where it is required that the domain contains the empty set (β β π). (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ dom πΉ = (π Γ π) & β’ ((π΄ β π β§ π΅ β π) β ((π΄πΉπ΅)) β π) & β’ ((π΄πΉπ΅)) β V β β’ ((π΄πΉπ΅)) β π | ||
Theorem | ndmaovrcl 46211 | Reverse closure law, in contrast to ndmovrcl 7597 where it is required that the operation's domain doesn't contain the empty set (Β¬ β β π), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ dom πΉ = (π Γ π) β β’ ( ((π΄πΉπ΅)) β π β (π΄ β π β§ π΅ β π)) | ||
Theorem | ndmaovcom 46212 | Any operation is commutative outside its domain, analogous to ndmovcom 7598. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ dom πΉ = (π Γ π) β β’ (Β¬ (π΄ β π β§ π΅ β π) β ((π΄πΉπ΅)) = ((π΅πΉπ΄)) ) | ||
Theorem | ndmaovass 46213 | Any operation is associative outside its domain. In contrast to ndmovass 7599 where it is required that the operation's domain doesn't contain the empty set (Β¬ β β π), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ dom πΉ = (π Γ π) β β’ (Β¬ (π΄ β π β§ π΅ β π β§ πΆ β π) β (( ((π΄πΉπ΅)) πΉπΆ)) = ((π΄πΉ ((π΅πΉπΆ)) )) ) | ||
Theorem | ndmaovdistr 46214 | Any operation is distributive outside its domain. In contrast to ndmovdistr 7600 where it is required that the operation's domain doesn't contain the empty set (Β¬ β β π), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ dom πΉ = (π Γ π) & β’ dom πΊ = (π Γ π) β β’ (Β¬ (π΄ β π β§ π΅ β π β§ πΆ β π) β ((π΄πΊ ((π΅πΉπΆ)) )) = (( ((π΄πΊπ΅)) πΉ ((π΄πΊπΆ)) )) ) | ||
In the following, a second approach is followed to define function values alternately to df-afv 46127. 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 46126. 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 46216) 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 46218). 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 46231, 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 46219). 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 46267), but (πΉβπ΄) = β β (πΉ''''π΄) β ran πΉ is not generally valid, see afv2fv0 46272. The alternate definition, however, corresponds to the current definition ((πΉβπ΄) = (πΉ''''π΄)) if the function πΉ is defined at π΄ (see dfatafv2eqfv 46268). With this definition the following intuitive equivalence holds: (πΉ defAt π΄ β (πΉ''''π΄) β ran πΉ), see dfatafv2rnb 46234. 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 46223, fveq2 6891-> afv2eq2 46224, nffv 6901-> nfafv2 46225, csbfv12 6939-> csbafv212g , rlimdm 15500-> rlimdmafv2 46265, tz6.12-1 6914-> tz6.12-1-afv2 46248, fveu 6880-> afv2eu 46245. Six theorems proved by directly using df-fv 6551 are within a mathbox (fvsb 43514, uncov 36773) or not used (rlimdmafv 46184, avril1 29984) or experimental (dfafv2 46139, dfafv22 46266). 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 46220 resp. afv2ex 46221). 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 46246). 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 46246 can be used instead of fvres 6910. * tz6.12-2 6879 (-> tz6.12-2-afv2 46244): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6883 (-> afv2prc 46233), used in 193 proofs, ** tz6.12i 6919 (-> tz6.12i-afv2 46250), used - indirectly via fvbr0 6920 and fvrn0 6921 - in 19 proofs, and in fvclss 7243 used in fvclex 7949 used in fvresex 7950 (which is not used!) and in dcomex 10446 (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 35222, setrec2lem1 47826 (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 43963), by shftval 15026 (itself used in 11 proofs), by dffv5 35201 (mathbox) and by fvco2 6988 (-> afv2co2 46264). * fvopab5 7030: used only by ajval 30382 (not used) and by adjval 31411, which is used in adjval2 31412 (not used) and in adjbdln 31604 (used in 7 proofs). * zsum 15669: used (via isum 15670, sum0 15672, sumss 15675 and fsumsers 15679) in 76 proofs. * isumshft 15790: used in pserdv2 26179 (used in logtayl 26405, binomcxplemdvsum 43417) , eftlub 16057 (used in 4 proofs), binomcxplemnotnn0 43418 (used in binomcxp 43419 only) and logtayl 26405 (used in 4 proofs). * ovtpos 8230: used in 16 proofs. * zprod 15886: used in 3 proofs: iprod 15887, zprodn0 15888 and prodss 15896 * iprodclim3 15949: 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 15669, isumshft 15790, ovtpos 8230 and zprod 15886 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 46215 | 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 46124. |
class (πΉ''''π΄) | ||
Definition | df-afv2 46216* | 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 46217* | 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 46218 | 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 46219 | 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 46220 | 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 46221 | 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 46222 | 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 46223 | 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 46224 | 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 46225 | 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 46134 cannot be shown easily. (Contributed by AV, 4-Sep-2022.) |
β’ β²π₯πΉ & β’ β²π₯π΄ β β’ β²π₯(πΉ''''π΄) | ||
Theorem | csbafv212g 46226 | 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 7454. (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 46227 | 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 46228 | 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 46229 | 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 46230 | 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 46231 | 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 46232 | 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 46233 | 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 46234 | 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 46235 | 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 46236 | 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 46237 | 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 46238 | 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 46239 | 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 46240 | 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 46241 | 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 46242 | 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 46243 | 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 46244* | 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 46245* | 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 46246 | 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 46247* | 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 46248* | 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 46249* | 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 46250 | 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 46251 | 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 46252 | 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 46220). (Contributed by AV, 6-Sep-2022.) |
β’ ((πΉ defAt π΄ β§ π΅ β π) β ((πΉ''''π΄) = π΅ β π΄πΉπ΅)) | ||
Theorem | dfatopafv2b 46253 | 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 46254 | 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 46255 | 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 46256 | 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 46257 | 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 46258 | 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 46259 | 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 46260* | 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 46261 | 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 46262* | Lemma for dfatco 46263. (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 46263 | 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 46264 | 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 46265 | Two ways to express that a function has a limit, analogous to rlimdm 15500. (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 46266 | 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 46267 | 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 46268 | 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 46269 | 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 46270 | 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 46271 | 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 46272 | 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 46273 | 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 46274 | 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 46275 | 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 46276 | 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 46277 | 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 46278 | Extend wff notation to include the 'not elemet of' relation. |
class _β | ||
Definition | df-nelbr 46279* | 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 46280 | 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 46281 | 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 46282 | 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 46283 | 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 46284 | 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 46285* | 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 46286* | 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 46287 | 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 46288 | 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 46289 | 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 46290* | 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 46291 | 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 46292* | 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 46293* | 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 46294* | 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 46295* | 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 46296* | 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 46297* | 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 46298* | 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 46299* | 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 46300* | 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} |
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