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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fnafvelrn 45101 | A function's value belongs to its range, analogous to fnfvelrn 7027. (Contributed by Alexander van der Vekens, 25-May-2017.) |
β’ ((πΉ Fn π΄ β§ π΅ β π΄) β (πΉ'''π΅) β ran πΉ) | ||
Theorem | fafvelcdm 45102 | A function's value belongs to its codomain, analogous to ffvelcdm 7028. (Contributed by Alexander van der Vekens, 25-May-2017.) |
β’ ((πΉ:π΄βΆπ΅ β§ πΆ β π΄) β (πΉ'''πΆ) β π΅) | ||
Theorem | ffnafv 45103* | A function maps to a class to which all values belong, analogous to ffnfv 7061. (Contributed by Alexander van der Vekens, 25-May-2017.) |
β’ (πΉ:π΄βΆπ΅ β (πΉ Fn π΄ β§ βπ₯ β π΄ (πΉ'''π₯) β π΅)) | ||
Theorem | afvres 45104 | The value of a restricted function, analogous to fvres 6857. (Contributed by Alexander van der Vekens, 22-Jul-2017.) |
β’ (π΄ β π΅ β ((πΉ βΎ π΅)'''π΄) = (πΉ'''π΄)) | ||
Theorem | tz6.12-afv 45105* | Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 6863. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
β’ ((β¨π΄, π¦β© β πΉ β§ β!π¦β¨π΄, π¦β© β πΉ) β (πΉ'''π΄) = π¦) | ||
Theorem | tz6.12-1-afv 45106* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 6861. (Contributed by Alexander van der Vekens, 29-Nov-2017.) |
β’ ((π΄πΉπ¦ β§ β!π¦ π΄πΉπ¦) β (πΉ'''π΄) = π¦) | ||
Theorem | dmfcoafv 45107 | Domains of a function composition, analogous to dmfco 6933. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
β’ ((Fun πΊ β§ π΄ β dom πΊ) β (π΄ β dom (πΉ β πΊ) β (πΊ'''π΄) β dom πΉ)) | ||
Theorem | afvco2 45108 | Value of a function composition, analogous to fvco2 6934. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
β’ ((πΊ Fn π΄ β§ π β π΄) β ((πΉ β πΊ)'''π) = (πΉ'''(πΊ'''π))) | ||
Theorem | rlimdmafv 45109 | Two ways to express that a function has a limit, analogous to rlimdm 15368. (Contributed by Alexander van der Vekens, 27-Nov-2017.) |
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 | aoveq123d 45110 | Equality deduction for operation value, analogous to oveq123d 7371. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ (π β πΉ = πΊ) & β’ (π β π΄ = π΅) & β’ (π β πΆ = π·) β β’ (π β ((π΄πΉπΆ)) = ((π΅πΊπ·)) ) | ||
Theorem | nfaov 45111 | Bound-variable hypothesis builder for operation value, analogous to nfov 7380. To prove a deduction version of this analogous to nfovd 7379 is not quickly possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of alternative operation values is based on are not available (see nfafv 45068). (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ β²π₯π΄ & β’ β²π₯πΉ & β’ β²π₯π΅ β β’ β²π₯ ((π΄πΉπ΅)) | ||
Theorem | csbaovg 45112 | Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ (π΄ β π· β β¦π΄ / π₯β¦ ((π΅πΉπΆ)) = ((β¦π΄ / π₯β¦π΅β¦π΄ / π₯β¦πΉβ¦π΄ / π₯β¦πΆ)) ) | ||
Theorem | aovfundmoveq 45113 | If a class is a function restricted to an ordered pair of its domain, then the value of the operation on this pair is equal for both definitions. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ (πΉ defAt β¨π΄, π΅β© β ((π΄πΉπ΅)) = (π΄πΉπ΅)) | ||
Theorem | aovnfundmuv 45114 | If an ordered pair is not in the domain of a class or the class is not a function restricted to the ordered pair, then the operation value for this pair is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ (Β¬ πΉ defAt β¨π΄, π΅β© β ((π΄πΉπ΅)) = V) | ||
Theorem | ndmaov 45115 | The value of an operation outside its domain, analogous to ndmafv 45072. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ (Β¬ β¨π΄, π΅β© β dom πΉ β ((π΄πΉπ΅)) = V) | ||
Theorem | ndmaovg 45116 | The value of an operation outside its domain, analogous to ndmovg 7530. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ ((dom πΉ = (π Γ π) β§ Β¬ (π΄ β π β§ π΅ β π)) β ((π΄πΉπ΅)) = V) | ||
Theorem | aovvdm 45117 | If the operation value of a class for an ordered pair is a set, the ordered pair is contained in the domain of the class. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ ( ((π΄πΉπ΅)) β πΆ β β¨π΄, π΅β© β dom πΉ) | ||
Theorem | nfunsnaov 45118 | If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ (Β¬ Fun (πΉ βΎ {β¨π΄, π΅β©}) β ((π΄πΉπ΅)) = V) | ||
Theorem | aovvfunressn 45119 | If the operation value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ ( ((π΄πΉπ΅)) β πΆ β Fun (πΉ βΎ {β¨π΄, π΅β©})) | ||
Theorem | aovprc 45120 | The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7388. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ Rel dom πΉ β β’ (Β¬ (π΄ β V β§ π΅ β V) β ((π΄πΉπ΅)) = V) | ||
Theorem | aovrcl 45121 | Reverse closure for an operation value, analogous to afvvv 45077. In contrast to ovrcl 7391, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.) |
β’ Rel dom πΉ β β’ ( ((π΄πΉπ΅)) β πΆ β (π΄ β V β§ π΅ β V)) | ||
Theorem | aovpcov0 45122 | 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.) |
β’ ( ((π΄πΉπ΅)) = V β (π΄πΉπ΅) = β ) | ||
Theorem | aovnuoveq 45123 | 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.) |
β’ ( ((π΄πΉπ΅)) β V β ((π΄πΉπ΅)) = (π΄πΉπ΅)) | ||
Theorem | aovvoveq 45124 | 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.) |
β’ ( ((π΄πΉπ΅)) β πΆ β ((π΄πΉπ΅)) = (π΄πΉπ΅)) | ||
Theorem | aov0ov0 45125 | 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.) |
β’ ( ((π΄πΉπ΅)) = β β (π΄πΉπ΅) = β ) | ||
Theorem | aovovn0oveq 45126 | 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 45127 | 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 45128 | 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 45129* | A frequently used special case of rspc2ev 3591 for operation values, analogous to rspceov 7397. (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 45130 | Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6892. (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 45131* | An operation maps to a class to which all values belong, analogous to ffnov 7476. (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 45132 | Closure law for an operation, analogous to fovcl 7477. (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 45133* | Value of a function given by the maps-to notation, analogous to ovmpt4g 7495. (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 45134* | Domain of closure of an operation. In contrast to oprssdm 7528, 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 45135 | The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7532 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 45136 | Reverse closure law, in contrast to ndmovrcl 7533 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 45137 | Any operation is commutative outside its domain, analogous to ndmovcom 7534. (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 45138 | Any operation is associative outside its domain. In contrast to ndmovass 7535 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 45139 | Any operation is distributive outside its domain. In contrast to ndmovdistr 7536 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 45052. The current definition of the value (πΉβπ΄) of a function πΉ at an argument π΄ (see df-fv 6500) assures that this value is always a set, see fex 7171. 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 6873 and fvprc 6830). "(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 45051. 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 6874). To avoid such an ambiguity, an alternative definition (πΉ''''π΄) (see df-afv2 45141) 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 45143). 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 45156, 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 45144). 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 45192), but (πΉβπ΄) = β β (πΉ''''π΄) β ran πΉ is not generally valid, see afv2fv0 45197. The alternate definition, however, corresponds to the current definition ((πΉβπ΄) = (πΉ''''π΄)) if the function πΉ is defined at π΄ (see dfatafv2eqfv 45193). With this definition the following intuitive equivalence holds: (πΉ defAt π΄ β (πΉ''''π΄) β ran πΉ), see dfatafv2rnb 45159. 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 6500 of (πΉβπ΄), we see that analogues for the following 7 theorems can be proven using the alternative definition: fveq1 6837-> afv2eq1 45148, fveq2 6838-> afv2eq2 45149, nffv 6848-> nfafv2 45150, csbfv12 6886-> csbafv212g , rlimdm 15368-> rlimdmafv2 45190, tz6.12-1 6861-> tz6.12-1-afv2 45173, fveu 6827-> afv2eu 45170. Six theorems proved by directly using df-fv 6500 are within a mathbox (fvsb 42465, uncov 35945) or not used (rlimdmafv 45109, avril1 29193) or experimental (dfafv2 45064, dfafv22 45191). However, the remaining 11 theorems proved by directly using df-fv 6500 are used more or less often: * fvex 6851: 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 45145 resp. afv2ex 45146). All of these 1600 proofs have to be checked if one of these two theorems can be used instead of fvex 6851. * fvres 6857: used in about 400 proofs : Only if the function is defined at the argument, an analog theorem can be proven (afv2res 45171). 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 45171 can be used instead of fvres 6857. * tz6.12-2 6826 (-> tz6.12-2-afv2 45169): root theorem of many theorems which have not a strict analogue, and which are used many times: ** fvprc 6830 (-> afv2prc 45158), used in 193 proofs, ** tz6.12i 6866 (-> tz6.12i-afv2 45175), used - indirectly via fvbr0 6867 and fvrn0 6868 - in 19 proofs, and in fvclss 7184 used in fvclex 7882 used in fvresex 7883 (which is not used!) and in dcomex 10317 (used in 4 proofs), ** ndmfv 6873 (-> ndmafv2nrn ), used in 124 proofs ** nfunsn 6880 (-> nfunsnafv2 ), used by fvfundmfvn0 6881 (used in 3 proofs), and dffv2 6932 (not used) ** funpartfv 34416, setrec2lem1 46838 (mathboxes) * fv2 6833: only used by elfv 6836, which is only used by fv3 6856, which is not used. * dffv3 6834 (-> dfafv23 ): used by dffv4 6835 (the previous "df-fv"), which now is only used in mathboxes (csbfv12gALTVD 42914), by shftval 14893 (itself used in 11 proofs), by dffv5 34395 (mathbox) and by fvco2 6934 (-> afv2co2 45189). * fvopab5 6976: used only by ajval 29589 (not used) and by adjval 30618, which is used in adjval2 30619 (not used) and in adjbdln 30811 (used in 7 proofs). * zsum 15538: used (via isum 15539, sum0 15541, sumss 15544 and fsumsers 15548) in 76 proofs. * isumshft 15659: used in pserdv2 25711 (used in logtayl 25937, binomcxplemdvsum 42368) , eftlub 15926 (used in 4 proofs), binomcxplemnotnn0 42369 (used in binomcxp 42370 only) and logtayl 25937 (used in 4 proofs). * ovtpos 8140: used in 16 proofs. * zprod 15755: used in 3 proofs: iprod 15756, zprodn0 15757 and prodss 15765 * iprodclim3 15818: 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 6833, dffv3 6834, fvopab5 6976, zsum 15538, isumshft 15659, ovtpos 8140 and zprod 15755 are not critical or are, hopefully, also valid for the alternative definition, fvex 6851, fvres 6857 and tz6.12-2 6826 (and the theorems based on them) are essential for the current definition of function values. | ||
Syntax | cafv2 45140 | 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 45049. |
class (πΉ''''π΄) | ||
Definition | df-afv2 45141* | 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 6500, and especially ndmfv 6873), (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 45142* | 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 45143 | 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 45144 | 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 45145 | 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 45146 | 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 45147 | Equality deduction for function value, analogous to fveq12d 6845. (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 45148 | Equality theorem for function value, analogous to fveq1 6837. (Contributed by AV, 4-Sep-2022.) |
β’ (πΉ = πΊ β (πΉ''''π΄) = (πΊ''''π΄)) | ||
Theorem | afv2eq2 45149 | Equality theorem for function value, analogous to fveq2 6838. (Contributed by AV, 4-Sep-2022.) |
β’ (π΄ = π΅ β (πΉ''''π΄) = (πΉ''''π΅)) | ||
Theorem | nfafv2 45150 | Bound-variable hypothesis builder for function value, analogous to nffv 6848. To prove a deduction version of this analogous to nffvd 6850 is not easily possible because a deduction version of nfdfat 45059 cannot be shown easily. (Contributed by AV, 4-Sep-2022.) |
β’ β²π₯πΉ & β’ β²π₯π΄ β β’ β²π₯(πΉ''''π΄) | ||
Theorem | csbafv212g 45151 | Move class substitution in and out of a function value, analogous to csbfv12 6886, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7392. (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 45152 | 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 45153 | 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 45154 | The value of a class outside its domain is not in the range, compare with ndmfv 6873. (Contributed by AV, 2-Sep-2022.) |
β’ (Β¬ π΄ β dom πΉ β (πΉ''''π΄) β ran πΉ) | ||
Theorem | funressndmafv2rn 45155 | 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 45156 | 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 45157 | If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6880. (Contributed by AV, 2-Sep-2022.) |
β’ (Β¬ Fun (πΉ βΎ {π΄}) β (πΉ''''π΄) β ran πΉ) | ||
Theorem | afv2prc 45158 | A function's value at a proper class is not defined, compare with fvprc 6830. (Contributed by AV, 5-Sep-2022.) |
β’ (Β¬ π΄ β V β (πΉ''''π΄) β ran πΉ) | ||
Theorem | dfatafv2rnb 45159 | 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 45160 | 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 45161 | 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 45162 | 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 45163 | An alternate function value belongs to the range of the function, analogous to fvelrn 7023. (Contributed by AV, 3-Sep-2022.) |
β’ ((Fun πΉ β§ π΄ β dom πΉ) β (πΉ''''π΄) β ran πΉ) | ||
Theorem | afv20defat 45164 | 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 45165 | An alternate function value belongs to the range of the function, analogous to fnfvelrn 7027. (Contributed by AV, 2-Sep-2022.) |
β’ ((πΉ Fn π΄ β§ π΅ β π΄) β (πΉ''''π΅) β ran πΉ) | ||
Theorem | fafv2elcdm 45166 | An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7028. (Contributed by AV, 2-Sep-2022.) |
β’ ((πΉ:π΄βΆπ΅ β§ πΆ β π΄) β (πΉ''''πΆ) β π΅) | ||
Theorem | fafv2elrnb 45167 | 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 45168 | 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 45169* | Function value when πΉ is (locally) not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27, analogous to tz6.12-2 6826. (Contributed by AV, 5-Sep-2022.) |
β’ (Β¬ β!π₯ π΄πΉπ₯ β (πΉ''''π΄) β ran πΉ) | ||
Theorem | afv2eu 45170* | The value of a function at a unique point, analogous to fveu 6827. (Contributed by AV, 5-Sep-2022.) |
β’ (β!π₯ π΄πΉπ₯ β (πΉ''''π΄) = βͺ {π₯ β£ π΄πΉπ₯}) | ||
Theorem | afv2res 45171 | The value of a restricted function for an argument at which the function is defined. Analog to fvres 6857. (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 45172* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6863. (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 45173* | Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6861. (Contributed by AV, 5-Sep-2022.) |
β’ ((π΄πΉπ¦ β§ β!π¦ π΄πΉπ¦) β (πΉ''''π΄) = π¦) | ||
Theorem | tz6.12c-afv2 45174* | Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6860. (Contributed by AV, 5-Sep-2022.) |
β’ (β!π¦ π΄πΉπ¦ β ((πΉ''''π΄) = π¦ β π΄πΉπ¦)) | ||
Theorem | tz6.12i-afv2 45175 | Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6866. (Contributed by AV, 5-Sep-2022.) |
β’ (π΅ β ran πΉ β ((πΉ''''π΄) = π΅ β π΄πΉπ΅)) | ||
Theorem | funressnbrafv2 45176 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6889. (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 45177 | Equivalence of function value and binary relation, analogous to fnbrfvb 6891 or funbrfvb 6893. 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 45145). (Contributed by AV, 6-Sep-2022.) |
β’ ((πΉ defAt π΄ β§ π΅ β π) β ((πΉ''''π΄) = π΅ β π΄πΉπ΅)) | ||
Theorem | dfatopafv2b 45178 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6892 or funopfvb 6894. (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 45179 | The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6889. (Contributed by AV, 6-Sep-2022.) |
β’ (Fun πΉ β (π΄πΉπ΅ β (πΉ''''π΄) = π΅)) | ||
Theorem | fnbrafv2b 45180 | Equivalence of function value and binary relation, analogous to fnbrfvb 6891. (Contributed by AV, 6-Sep-2022.) |
β’ ((πΉ Fn π΄ β§ π΅ β π΄) β ((πΉ''''π΅) = πΆ β π΅πΉπΆ)) | ||
Theorem | fnopafv2b 45181 | Equivalence of function value and ordered pair membership, analogous to fnopfvb 6892. (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 45182 | Equivalence of function value and binary relation, analogous to funbrfvb 6893. (Contributed by AV, 6-Sep-2022.) |
β’ ((Fun πΉ β§ π΄ β dom πΉ) β ((πΉ''''π΄) = π΅ β π΄πΉπ΅)) | ||
Theorem | funopafv2b 45183 | Equivalence of function value and ordered pair membership, analogous to funopfvb 6894. (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 45184 | Singleton of function value, analogous to fnsnfv 6916. (Contributed by AV, 7-Sep-2022.) |
β’ (πΉ defAt π΄ β {(πΉ''''π΄)} = (πΉ β {π΄})) | ||
Theorem | dfafv23 45185* | A definition of function value in terms of iota, analogous to dffv3 6834. (Contributed by AV, 6-Sep-2022.) |
β’ (πΉ defAt π΄ β (πΉ''''π΄) = (β©π₯π₯ β (πΉ β {π΄}))) | ||
Theorem | dfatdmfcoafv2 45186 | Domain of a function composition, analogous to dmfco 6933. (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 45187* | Lemma for dfatco 45188. (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 45188 | 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 45189 | Value of a function composition, analogous to fvco2 6934. (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 45190 | Two ways to express that a function has a limit, analogous to rlimdm 15368. (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 45191 | 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 45192 | 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 45193 | 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 45194 | 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 45195 | 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 45196 | 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 45197 | 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 45198 | 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 45199 | 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 45200 | 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))) |
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