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Theorem List for Metamath Proof Explorer - 46001-46100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.45.5  Alternative definitions of function values (2)

In the following, a second approach is followed to define function values alternately to df-afv 45913.

The current definition of the value (πΉβ€˜π΄) of a function 𝐹 at an argument 𝐴 (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 𝐹 and 𝐴, and evaluates to the empty set when it is not meaningful (as shown by ndmfv 6926 and fvprc 6883). "(πΉβ€˜π΄) is meaningful" means "the class 𝐹 regarded as function is defined at the argument 𝐴" in this context. This is also expressed by 𝐹 defAt 𝐴, see df-dfat 45912. In the theory of partial functions, it is a common case that 𝐹 is not defined at 𝐴.

Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (πΉβ€˜π΄) = βˆ… alone it cannot be decided/derived whether (πΉβ€˜π΄) is meaningful (𝐹 is actually a function which is defined for 𝐴 and really has the function value βˆ… at 𝐴) or not. Therefore, additional assumptions are required, such as βˆ… βˆ‰ ran 𝐹, βˆ… ∈ ran 𝐹, 𝐹 defAt 𝐴, or Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6927).

To avoid such an ambiguity, an alternative definition (𝐹''''𝐴) (see df-afv2 46002) would be possible which evaluates to a set not belonging to the range of 𝐹 ((𝐹''''𝐴) = 𝒫 βˆͺ ran 𝐹) if it is not meaningful (see ndfatafv2 46004). We say "(𝐹''''𝐴) is not defined (or undefined)" if (𝐹''''𝐴) is not in the range of 𝐹 ((𝐹''''𝐴) βˆ‰ ran 𝐹). Because of afv2ndefb 46017, this is equivalent to ((𝐹''''𝐴) = 𝒫 βˆͺ ran 𝐹. If (𝐹''''𝐴) is in the range of 𝐹 ((𝐹''''𝐴) ∈ ran 𝐹), we say that "(𝐹''''𝐴) is defined".

If ran 𝐹 is a set, we can use the symbol Undef to express that (𝐹''''𝐴) is not defined: (𝐹''''𝐴) = (Undefβ€˜ran 𝐹) (see ndfatafv2undef 46005). We could have used this symbol directly to define the alternate value of a function, which would have the advantage that (𝐹''''𝐴) would always be a set. But first this symbol is defined using the original function value, which would not make it possible to replace the original definition by the alternate definition, and second we would have to assume that ran 𝐹 ∈ V in most of the theorems.

To summarize, that means (𝐹''''𝐴) βˆ‰ ran 𝐹 β†’ (πΉβ€˜π΄) = βˆ… (see afv2ndeffv0 46053), but (πΉβ€˜π΄) = βˆ… β†’ (𝐹''''𝐴) βˆ‰ ran 𝐹 is not generally valid, see afv2fv0 46058.

The alternate definition, however, corresponds to the current definition ((πΉβ€˜π΄) = (𝐹''''𝐴)) if the function 𝐹 is defined at 𝐴 (see dfatafv2eqfv 46054).

With this definition the following intuitive equivalence holds: (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹), see dfatafv2rnb 46020.

An interesting question would be if (πΉβ€˜π΄) could be replaced by (𝐹'''𝐴) in most of the theorems based on function values. If we look at the (currently 24) proofs using the definition df-fv 6551 of (πΉβ€˜π΄), 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 (𝐹 β†Ύ 𝐡) is mostly different from the range of 𝐹, and therefore also the "undefined" values are different. All of these 400 proofs have to be checked if afv2res 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.

 
Syntaxcafv2 46001 Extend the definition of a class to include the alternate function value. Read: "the value of 𝐹 at 𝐴 " or "𝐹 of 𝐴". For using several apostrophes as a symbol see comment for cafv 45910.
class (𝐹''''𝐴)
 
Definitiondf-afv2 46002* Alternate definition of the value of a function, (𝐹''''𝐴), also known as function application (and called "alternate function value" in the following). In contrast to (πΉβ€˜π΄) = βˆ… (see comment of df-fv 6551, and especially ndmfv 6926), (𝐹''''𝐴) is guaranteed not to be in the range of 𝐹 if 𝐹 is not defined at 𝐴 (whereas βˆ… can be a member of ran 𝐹). (Contributed by AV, 2-Sep-2022.)
(𝐹''''𝐴) = if(𝐹 defAt 𝐴, (β„©π‘₯𝐴𝐹π‘₯), 𝒫 βˆͺ ran 𝐹)
 
Theoremdfatafv2iota 46003* If a function is defined at a class 𝐴 the alternate function value at 𝐴 is the unique value assigned to 𝐴 by the function (analogously to (πΉβ€˜π΄)). (Contributed by AV, 2-Sep-2022.)
(𝐹 defAt 𝐴 β†’ (𝐹''''𝐴) = (β„©π‘₯𝐴𝐹π‘₯))
 
Theoremndfatafv2 46004 The alternate function value at a class 𝐴 if the function is not defined at this set 𝐴. (Contributed by AV, 2-Sep-2022.)
(Β¬ 𝐹 defAt 𝐴 β†’ (𝐹''''𝐴) = 𝒫 βˆͺ ran 𝐹)
 
Theoremndfatafv2undef 46005 The alternate function value at a class 𝐴 is undefined if the function, whose range is a set, is not defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
((ran 𝐹 ∈ 𝑉 ∧ Β¬ 𝐹 defAt 𝐴) β†’ (𝐹''''𝐴) = (Undefβ€˜ran 𝐹))
 
Theoremdfatafv2ex 46006 The alternate function value at a class 𝐴 is always a set if the function/class 𝐹 is defined at 𝐴. (Contributed by AV, 6-Sep-2022.)
(𝐹 defAt 𝐴 β†’ (𝐹''''𝐴) ∈ V)
 
Theoremafv2ex 46007 The alternate function value is always a set if the range of the function is a set. (Contributed by AV, 2-Sep-2022.)
(ran 𝐹 ∈ 𝑉 β†’ (𝐹''''𝐴) ∈ V)
 
Theoremafv2eq12d 46008 Equality deduction for function value, analogous to fveq12d 6898. (Contributed by AV, 4-Sep-2022.)
(πœ‘ β†’ 𝐹 = 𝐺)    &   (πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ (𝐹''''𝐴) = (𝐺''''𝐡))
 
Theoremafv2eq1 46009 Equality theorem for function value, analogous to fveq1 6890. (Contributed by AV, 4-Sep-2022.)
(𝐹 = 𝐺 β†’ (𝐹''''𝐴) = (𝐺''''𝐴))
 
Theoremafv2eq2 46010 Equality theorem for function value, analogous to fveq2 6891. (Contributed by AV, 4-Sep-2022.)
(𝐴 = 𝐡 β†’ (𝐹''''𝐴) = (𝐹''''𝐡))
 
Theoremnfafv2 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„²π‘₯𝐹    &   β„²π‘₯𝐴    β‡’   β„²π‘₯(𝐹''''𝐴)
 
Theoremcsbafv212g 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.)
(𝐴 ∈ 𝑉 β†’ ⦋𝐴 / π‘₯⦌(𝐹''''𝐡) = (⦋𝐴 / π‘₯⦌𝐹''''⦋𝐴 / π‘₯⦌𝐡))
 
Theoremfexafv2ex 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.)
(𝐹 ∈ 𝑉 β†’ (𝐹''''𝐴) ∈ V)
 
Theoremndfatafv2nrn 46014 The alternate function value at a class 𝐴 at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022.)
(Β¬ 𝐹 defAt 𝐴 β†’ (𝐹''''𝐴) βˆ‰ ran 𝐹)
 
Theoremndmafv2nrn 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¬ 𝐴 ∈ dom 𝐹 β†’ (𝐹''''𝐴) βˆ‰ ran 𝐹)
 
Theoremfunressndmafv2rn 46016 The alternate function value at a class 𝐴 is defined, i.e., in the range of the function if the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
(𝐹 defAt 𝐴 β†’ (𝐹''''𝐴) ∈ ran 𝐹)
 
Theoremafv2ndefb 46017 Two ways to say that an alternate function value is not defined. (Contributed by AV, 5-Sep-2022.)
((𝐹''''𝐴) = 𝒫 βˆͺ ran 𝐹 ↔ (𝐹''''𝐴) βˆ‰ ran 𝐹)
 
Theoremnfunsnafv2 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¬ Fun (𝐹 β†Ύ {𝐴}) β†’ (𝐹''''𝐴) βˆ‰ ran 𝐹)
 
Theoremafv2prc 46019 A function's value at a proper class is not defined, compare with fvprc 6883. (Contributed by AV, 5-Sep-2022.)
(Β¬ 𝐴 ∈ V β†’ (𝐹''''𝐴) βˆ‰ ran 𝐹)
 
Theoremdfatafv2rnb 46020 The alternate function value at a class 𝐴 is defined, i.e. in the range of the function, iff the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
(𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
 
Theoremafv2orxorb 46021 If a set is in the range of a function, the alternate function value at a class 𝐴 equals this set or is not in the range of the function iff the alternate function value at the class 𝐴 either equals this set or is not in the range of the function. If 𝐡 βˆ‰ ran 𝐹, both disjuncts of the exclusive or can be true: (𝐹''''𝐴) = 𝐡 β†’ (𝐹''''𝐴) βˆ‰ ran 𝐹. (Contributed by AV, 11-Sep-2022.)
(𝐡 ∈ ran 𝐹 β†’ (((𝐹''''𝐴) = 𝐡 ∨ (𝐹''''𝐴) βˆ‰ ran 𝐹) ↔ ((𝐹''''𝐴) = 𝐡 ⊻ (𝐹''''𝐴) βˆ‰ ran 𝐹)))
 
Theoremdmafv2rnb 46022 The alternate function value at a class 𝐴 is defined, i.e., in the range of the function, iff 𝐴 is in the domain of the function. (Contributed by AV, 3-Sep-2022.)
(Fun (𝐹 β†Ύ {𝐴}) β†’ (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
 
Theoremfundmafv2rnb 46023 The alternate function value at a class 𝐴 is defined, i.e., in the range of the function iff 𝐴 is in the domain of the function. (Contributed by AV, 3-Sep-2022.)
(Fun 𝐹 β†’ (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
 
Theoremafv2elrn 46024 An alternate function value belongs to the range of the function, analogous to fvelrn 7078. (Contributed by AV, 3-Sep-2022.)
((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) β†’ (𝐹''''𝐴) ∈ ran 𝐹)
 
Theoremafv20defat 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.)
((𝐹''''𝐴) = βˆ… β†’ 𝐹 defAt 𝐴)
 
Theoremfnafv2elrn 46026 An alternate function value belongs to the range of the function, analogous to fnfvelrn 7082. (Contributed by AV, 2-Sep-2022.)
((𝐹 Fn 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ (𝐹''''𝐡) ∈ ran 𝐹)
 
Theoremfafv2elcdm 46027 An alternate function value belongs to the codomain of the function, analogous to ffvelcdm 7083. (Contributed by AV, 2-Sep-2022.)
((𝐹:𝐴⟢𝐡 ∧ 𝐢 ∈ 𝐴) β†’ (𝐹''''𝐢) ∈ 𝐡)
 
Theoremfafv2elrnb 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.)
(𝐹:𝐴⟢𝐡 β†’ (𝐢 ∈ 𝐴 ↔ (𝐹''''𝐢) ∈ ran 𝐹))
 
Theoremfcdmvafv2v 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.)
((𝐹:𝐴⟢𝐡 ∧ 𝐡 ∈ 𝑉) β†’ (𝐹''''𝐢) ∈ V)
 
Theoremtz6.12-2-afv2 46030* Function value when 𝐹 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¬ βˆƒ!π‘₯ 𝐴𝐹π‘₯ β†’ (𝐹''''𝐴) βˆ‰ ran 𝐹)
 
Theoremafv2eu 46031* The value of a function at a unique point, analogous to fveu 6880. (Contributed by AV, 5-Sep-2022.)
(βˆƒ!π‘₯ 𝐴𝐹π‘₯ β†’ (𝐹''''𝐴) = βˆͺ {π‘₯ ∣ 𝐴𝐹π‘₯})
 
Theoremafv2res 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.)
((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐡) β†’ ((𝐹 β†Ύ 𝐡)''''𝐴) = (𝐹''''𝐴))
 
Theoremtz6.12-afv2 46033* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12 6916. (Contributed by AV, 5-Sep-2022.)
((⟨𝐴, π‘¦βŸ© ∈ 𝐹 ∧ βˆƒ!π‘¦βŸ¨π΄, π‘¦βŸ© ∈ 𝐹) β†’ (𝐹''''𝐴) = 𝑦)
 
Theoremtz6.12-1-afv2 46034* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27), analogous to tz6.12-1 6914. (Contributed by AV, 5-Sep-2022.)
((𝐴𝐹𝑦 ∧ βˆƒ!𝑦 𝐴𝐹𝑦) β†’ (𝐹''''𝐴) = 𝑦)
 
Theoremtz6.12c-afv2 46035* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12c 6913. (Contributed by AV, 5-Sep-2022.)
(βˆƒ!𝑦 𝐴𝐹𝑦 β†’ ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
 
Theoremtz6.12i-afv2 46036 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6919. (Contributed by AV, 5-Sep-2022.)
(𝐡 ∈ ran 𝐹 β†’ ((𝐹''''𝐴) = 𝐡 β†’ 𝐴𝐹𝐡))
 
Theoremfunressnbrafv2 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.)
(((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) ∧ Fun (𝐹 β†Ύ {𝐴})) β†’ (𝐴𝐹𝐡 β†’ (𝐹''''𝐴) = 𝐡))
 
Theoremdfatbrafv2b 46038 Equivalence of function value and binary relation, analogous to fnbrfvb 6944 or funbrfvb 6946. 𝐡 ∈ V is required, because otherwise 𝐴𝐹𝐡 ↔ βˆ… ∈ 𝐹 can be true, but (𝐹''''𝐴) = 𝐡 is always false (because of dfatafv2ex 46006). (Contributed by AV, 6-Sep-2022.)
((𝐹 defAt 𝐴 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐹''''𝐴) = 𝐡 ↔ 𝐴𝐹𝐡))
 
Theoremdfatopafv2b 46039 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6945 or funopfvb 6947. (Contributed by AV, 6-Sep-2022.)
((𝐹 defAt 𝐴 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐹''''𝐴) = 𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ 𝐹))
 
Theoremfunbrafv2 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.)
(Fun 𝐹 β†’ (𝐴𝐹𝐡 β†’ (𝐹''''𝐴) = 𝐡))
 
Theoremfnbrafv2b 46041 Equivalence of function value and binary relation, analogous to fnbrfvb 6944. (Contributed by AV, 6-Sep-2022.)
((𝐹 Fn 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ ((𝐹''''𝐡) = 𝐢 ↔ 𝐡𝐹𝐢))
 
Theoremfnopafv2b 46042 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6945. (Contributed by AV, 6-Sep-2022.)
((𝐹 Fn 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ ((𝐹''''𝐡) = 𝐢 ↔ ⟨𝐡, 𝐢⟩ ∈ 𝐹))
 
Theoremfunbrafv22b 46043 Equivalence of function value and binary relation, analogous to funbrfvb 6946. (Contributed by AV, 6-Sep-2022.)
((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) β†’ ((𝐹''''𝐴) = 𝐡 ↔ 𝐴𝐹𝐡))
 
Theoremfunopafv2b 46044 Equivalence of function value and ordered pair membership, analogous to funopfvb 6947. (Contributed by AV, 6-Sep-2022.)
((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) β†’ ((𝐹''''𝐴) = 𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ 𝐹))
 
Theoremdfatsnafv2 46045 Singleton of function value, analogous to fnsnfv 6970. (Contributed by AV, 7-Sep-2022.)
(𝐹 defAt 𝐴 β†’ {(𝐹''''𝐴)} = (𝐹 β€œ {𝐴}))
 
Theoremdfafv23 46046* A definition of function value in terms of iota, analogous to dffv3 6887. (Contributed by AV, 6-Sep-2022.)
(𝐹 defAt 𝐴 β†’ (𝐹''''𝐴) = (β„©π‘₯π‘₯ ∈ (𝐹 β€œ {𝐴})))
 
Theoremdfatdmfcoafv2 46047 Domain of a function composition, analogous to dmfco 6987. (Contributed by AV, 7-Sep-2022.)
(𝐺 defAt 𝐴 β†’ (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))
 
Theoremdfatcolem 46048* Lemma for dfatco 46049. (Contributed by AV, 8-Sep-2022.)
((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) β†’ βˆƒ!𝑦 𝑋(𝐹 ∘ 𝐺)𝑦)
 
Theoremdfatco 46049 The predicate "defined at" for a function composition. (Contributed by AV, 8-Sep-2022.)
((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) β†’ (𝐹 ∘ 𝐺) defAt 𝑋)
 
Theoremafv2co2 46050 Value of a function composition, analogous to fvco2 6988. (Contributed by AV, 8-Sep-2022.)
((𝐺 defAt 𝑋 ∧ 𝐹 defAt (𝐺''''𝑋)) β†’ ((𝐹 ∘ 𝐺)''''𝑋) = (𝐹''''(𝐺''''𝑋)))
 
Theoremrlimdmafv2 46051 Two ways to express that a function has a limit, analogous to rlimdm 15497. (Contributed by AV, 5-Sep-2022.)
(πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ sup(𝐴, ℝ*, < ) = +∞)    β‡’   (πœ‘ β†’ (𝐹 ∈ dom β‡π‘Ÿ ↔ 𝐹 β‡π‘Ÿ ( β‡π‘Ÿ ''''𝐹)))
 
Theoremdfafv22 46052 Alternate definition of (𝐹''''𝐴) using (πΉβ€˜π΄) directly. (Contributed by AV, 3-Sep-2022.)
(𝐹''''𝐴) = if(𝐹 defAt 𝐴, (πΉβ€˜π΄), 𝒫 βˆͺ ran 𝐹)
 
Theoremafv2ndeffv0 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.)
((𝐹''''𝐴) βˆ‰ ran 𝐹 β†’ (πΉβ€˜π΄) = βˆ…)
 
Theoremdfatafv2eqfv 46054 If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.)
(𝐹 defAt 𝐴 β†’ (𝐹''''𝐴) = (πΉβ€˜π΄))
 
Theoremafv2rnfveq 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.)
((𝐹''''𝐴) ∈ ran 𝐹 β†’ (𝐹''''𝐴) = (πΉβ€˜π΄))
 
Theoremafv20fv0 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.)
((𝐹''''𝐴) = βˆ… β†’ (πΉβ€˜π΄) = βˆ…)
 
Theoremafv2fvn0fveq 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.)
((πΉβ€˜π΄) β‰  βˆ… β†’ (𝐹''''𝐴) = (πΉβ€˜π΄))
 
Theoremafv2fv0 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.)
((πΉβ€˜π΄) = βˆ… β†’ ((𝐹''''𝐴) = βˆ… ∨ (𝐹''''𝐴) βˆ‰ ran 𝐹))
 
Theoremafv2fv0b 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.)
((πΉβ€˜π΄) = βˆ… ↔ ((𝐹''''𝐴) = βˆ… ∨ (𝐹''''𝐴) βˆ‰ ran 𝐹))
 
Theoremafv2fv0xorb 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.)
(βˆ… ∈ ran 𝐹 β†’ ((πΉβ€˜π΄) = βˆ… ↔ ((𝐹''''𝐴) = βˆ… ⊻ (𝐹''''𝐴) βˆ‰ ran 𝐹)))
 
21.45.6  General auxiliary theorems (2)
 
21.45.6.1  Logical conjunction - extension
 
Theoreman4com24 46061 Rearrangement of 4 conjuncts: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
(((πœ‘ ∧ πœ“) ∧ (πœ’ ∧ πœƒ)) ↔ ((πœ‘ ∧ πœƒ) ∧ (πœ’ ∧ πœ“)))
 
21.45.6.2  Abbreviated conjunction and disjunction of three wff's - extension
 
Theorem3an4ancom24 46062 Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
(((πœ‘ ∧ πœ“ ∧ πœ’) ∧ πœƒ) ↔ ((πœ‘ ∧ πœƒ ∧ πœ’) ∧ πœ“))
 
Theorem4an21 46063 Rearrangement of 4 conjuncts with a triple conjunction. (Contributed by AV, 4-Mar-2022.)
(((πœ‘ ∧ πœ“) ∧ πœ’ ∧ πœƒ) ↔ (πœ“ ∧ (πœ‘ ∧ πœ’ ∧ πœƒ)))
 
21.45.6.3  Negated membership (alternative)
 
Syntaxcnelbr 46064 Extend wff notation to include the 'not elemet of' relation.
class _βˆ‰
 
Definitiondf-nelbr 46065* Define negated membership as binary relation. Analogous to df-eprel 5580 (the membership relation). (Contributed by AV, 26-Dec-2021.)
_βˆ‰ = {⟨π‘₯, π‘¦βŸ© ∣ Β¬ π‘₯ ∈ 𝑦}
 
Theoremdfnelbr2 46066 Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.)
_βˆ‰ = ((V Γ— V) βˆ– E )
 
Theoremnelbr 46067 The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 _βˆ‰ 𝐡 ↔ Β¬ 𝐴 ∈ 𝐡))
 
Theoremnelbrim 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 𝐴 is a proper class, then Β¬ 𝐴 ∈ 𝐡 would be true, but not 𝐴 _βˆ‰ 𝐡. (Contributed by AV, 26-Dec-2021.)
(𝐴 _βˆ‰ 𝐡 β†’ Β¬ 𝐴 ∈ 𝐡)
 
Theoremnelbrnel 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.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 _βˆ‰ 𝐡 ↔ 𝐴 βˆ‰ 𝐡))
 
Theoremnelbrnelim 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.)
(𝐴 _βˆ‰ 𝐡 β†’ 𝐴 βˆ‰ 𝐡)
 
21.45.6.4  The empty set - extension
 
Theoremralralimp 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.)
((πœ‘ ∧ 𝐴 β‰  βˆ…) β†’ (βˆ€π‘₯ ∈ 𝐴 ((πœ‘ β†’ (πœƒ ∨ 𝜏)) ∧ Β¬ πœƒ) β†’ 𝜏))
 
21.45.6.5  Indexed union and intersection - extension
 
TheoremotiunsndisjX 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.)
(𝐡 ∈ 𝑋 β†’ Disj π‘Ž ∈ 𝑉 βˆͺ 𝑐 ∈ π‘Š {βŸ¨π‘Ž, 𝐡, π‘βŸ©})
 
21.45.6.6  Functions - extension
 
Theoremfvifeq 46073 Equality of function values with conditional arguments, see also fvif 6907. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐴 = if(πœ‘, 𝐡, 𝐢) β†’ (πΉβ€˜π΄) = if(πœ‘, (πΉβ€˜π΅), (πΉβ€˜πΆ)))
 
Theoremrnfdmpr 46074 The range of a one-to-one function 𝐹 of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ π‘Š) β†’ (𝐹 Fn {𝑋, π‘Œ} β†’ ran 𝐹 = {(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)}))
 
Theoremimarnf1pr 46075 The image of the range of a function 𝐹 under a function 𝐸 if 𝐹 is a function from a pair into the domain of 𝐸. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ π‘Š) β†’ (((𝐹:{𝑋, π‘Œ}⟢dom 𝐸 ∧ 𝐸:dom πΈβŸΆπ‘…) ∧ ((πΈβ€˜(πΉβ€˜π‘‹)) = 𝐴 ∧ (πΈβ€˜(πΉβ€˜π‘Œ)) = 𝐡)) β†’ (𝐸 β€œ ran 𝐹) = {𝐴, 𝐡}))
 
Theoremfunop1 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.)
(βˆƒπ‘₯βˆƒπ‘¦ 𝐹 = ⟨π‘₯, π‘¦βŸ© β†’ (Fun 𝐹 ↔ βˆƒπ‘₯βˆƒπ‘¦ 𝐹 = {⟨π‘₯, π‘¦βŸ©}))
 
Theoremfun2dmnopgexmpl 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.)
(𝐺 = {⟨0, 1⟩, ⟨1, 1⟩} β†’ Β¬ 𝐺 ∈ (V Γ— V))
 
Theoremopabresex0d 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.)
((πœ‘ ∧ π‘₯𝑅𝑦) β†’ π‘₯ ∈ 𝐢)    &   ((πœ‘ ∧ π‘₯𝑅𝑦) β†’ πœƒ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ {𝑦 ∣ πœƒ} ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ π‘Š)    β‡’   (πœ‘ β†’ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯𝑅𝑦 ∧ πœ“)} ∈ V)
 
Theoremopabbrfex0d 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.)
((πœ‘ ∧ π‘₯𝑅𝑦) β†’ π‘₯ ∈ 𝐢)    &   ((πœ‘ ∧ π‘₯𝑅𝑦) β†’ πœƒ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ {𝑦 ∣ πœƒ} ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ π‘Š)    β‡’   (πœ‘ β†’ {⟨π‘₯, π‘¦βŸ© ∣ π‘₯𝑅𝑦} ∈ V)
 
Theoremopabresexd 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.)
((πœ‘ ∧ π‘₯𝑅𝑦) β†’ π‘₯ ∈ 𝐢)    &   ((πœ‘ ∧ π‘₯𝑅𝑦) β†’ 𝑦:𝐴⟢𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐴 ∈ π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ π‘Š)    β‡’   (πœ‘ β†’ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯𝑅𝑦 ∧ πœ“)} ∈ V)
 
Theoremopabbrfexd 46081* A collection of ordered pairs, the second component being a function, is a set. (Contributed by AV, 15-Jan-2021.)
((πœ‘ ∧ π‘₯𝑅𝑦) β†’ π‘₯ ∈ 𝐢)    &   ((πœ‘ ∧ π‘₯𝑅𝑦) β†’ 𝑦:𝐴⟢𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐴 ∈ π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ 𝐢 ∈ π‘Š)    β‡’   (πœ‘ β†’ {⟨π‘₯, π‘¦βŸ© ∣ π‘₯𝑅𝑦} ∈ V)
 
Theoremf1oresf1orab 46082* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 1-Aug-2022.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐢)    &   (πœ‘ β†’ 𝐹:𝐴–1-1-onto→𝐡)    &   (πœ‘ β†’ 𝐷 βŠ† 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐢) β†’ (πœ’ ↔ π‘₯ ∈ 𝐷))    β‡’   (πœ‘ β†’ (𝐹 β†Ύ 𝐷):𝐷–1-1-ontoβ†’{𝑦 ∈ 𝐡 ∣ πœ’})
 
Theoremf1oresf1o 46083* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
(πœ‘ β†’ 𝐹:𝐴–1-1-onto→𝐡)    &   (πœ‘ β†’ 𝐷 βŠ† 𝐴)    &   (πœ‘ β†’ (βˆƒπ‘₯ ∈ 𝐷 (πΉβ€˜π‘₯) = 𝑦 ↔ (𝑦 ∈ 𝐡 ∧ πœ’)))    β‡’   (πœ‘ β†’ (𝐹 β†Ύ 𝐷):𝐷–1-1-ontoβ†’{𝑦 ∈ 𝐡 ∣ πœ’})
 
Theoremf1oresf1o2 46084* Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
(πœ‘ β†’ 𝐹:𝐴–1-1-onto→𝐡)    &   (πœ‘ β†’ 𝐷 βŠ† 𝐴)    &   ((πœ‘ ∧ 𝑦 = (πΉβ€˜π‘₯)) β†’ (π‘₯ ∈ 𝐷 ↔ πœ’))    β‡’   (πœ‘ β†’ (𝐹 β†Ύ 𝐷):𝐷–1-1-ontoβ†’{𝑦 ∈ 𝐡 ∣ πœ’})
 
21.45.6.7  Maps-to notation - extension
 
Theoremfvmptrab 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.)
𝐹 = (π‘₯ ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ πœ‘})    &   (π‘₯ = 𝑋 β†’ (πœ‘ ↔ πœ“))    &   (π‘₯ = 𝑋 β†’ 𝑀 = 𝑁)    &   (𝑋 ∈ 𝑉 β†’ 𝑁 ∈ V)    &   (𝑋 βˆ‰ 𝑉 β†’ 𝑁 = βˆ…)    β‡’   (πΉβ€˜π‘‹) = {𝑦 ∈ 𝑁 ∣ πœ“}
 
Theoremfvmptrabdm 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.)
𝐹 = (π‘₯ ∈ 𝑉 ↦ {𝑦 ∈ (πΊβ€˜π‘Œ) ∣ πœ‘})    &   (π‘₯ = 𝑋 β†’ (πœ‘ ↔ πœ“))    &   (π‘Œ ∈ dom 𝐺 β†’ 𝑋 ∈ dom 𝐹)    β‡’   (πΉβ€˜π‘‹) = {𝑦 ∈ (πΊβ€˜π‘Œ) ∣ πœ“}
 
21.45.6.8  Subtraction - extension
 
Theoremcnambpcma 46087 ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ (((𝐴 βˆ’ 𝐡) + 𝐢) βˆ’ 𝐴) = (𝐢 βˆ’ 𝐡))
 
Theoremcnapbmcpd 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.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝐢 ∈ β„‚ ∧ 𝐷 ∈ β„‚)) β†’ (((𝐴 + 𝐡) βˆ’ 𝐢) + 𝐷) = (((𝐴 + 𝐷) + 𝐡) βˆ’ 𝐢))
 
Theoremaddsubeq0 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.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐴 + 𝐡) = (𝐴 βˆ’ 𝐡) ↔ 𝐡 = 0))
 
21.45.6.9  Ordering on reals (cont.) - extension
 
Theoremleaddsuble 46090 Addition and subtraction on one side of "less than or equal to". (Contributed by Alexander van der Vekens, 18-Mar-2018.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ (𝐡 ≀ 𝐢 ↔ ((𝐴 + 𝐡) βˆ’ 𝐢) ≀ 𝐴))
 
Theorem2leaddle2 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.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 < 𝐢 ∧ 𝐡 < 𝐢) β†’ (𝐴 + 𝐡) < (2 Β· 𝐢)))
 
Theoremltnltne 46092 Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 < 𝐡 ↔ (Β¬ 𝐡 < 𝐴 ∧ Β¬ 𝐡 = 𝐴)))
 
Theoremp1lep2 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.)
(𝑁 ∈ ℝ β†’ (𝑁 + 1) ≀ (𝑁 + 2))
 
Theoremltsubsubaddltsub 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.)
((𝐽 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ)) β†’ (𝐽 < ((𝐿 βˆ’ 𝑀) βˆ’ 𝑁) ↔ (𝐽 + 𝑀) < (𝐿 βˆ’ 𝑁)))
 
Theoremzm1nn 46095 An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
((𝑁 ∈ β„•0 ∧ 𝐿 ∈ β„€) β†’ ((𝐽 ∈ ℝ ∧ 0 ≀ 𝐽 ∧ 𝐽 < ((𝐿 βˆ’ 𝑁) βˆ’ 1)) β†’ (𝐿 βˆ’ 1) ∈ β„•))
 
21.45.6.10  Imaginary and complex number properties - extension
 
Theoremreaddcnnred 46096 The sum of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ (β„‚ βˆ– ℝ))    β‡’   (πœ‘ β†’ (𝐴 + 𝐡) βˆ‰ ℝ)
 
Theoremresubcnnred 46097 The difference of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ (β„‚ βˆ– ℝ))    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) βˆ‰ ℝ)
 
Theoremrecnmulnred 46098 The product of a real number and an imaginary number is not a real number. (Contributed by AV, 23-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ (β„‚ βˆ– ℝ))    &   (πœ‘ β†’ 𝐴 β‰  0)    β‡’   (πœ‘ β†’ (𝐴 Β· 𝐡) βˆ‰ ℝ)
 
Theoremcndivrenred 46099 The quotient of an imaginary number and a real number is not a real number. (Contributed by AV, 23-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ (β„‚ βˆ– ℝ))    &   (πœ‘ β†’ 𝐴 β‰  0)    β‡’   (πœ‘ β†’ (𝐡 / 𝐴) βˆ‰ ℝ)
 
Theoremsqrtnegnre 46100 The square root of a negative number is not a real number. (Contributed by AV, 28-Feb-2023.)
((𝑋 ∈ ℝ ∧ 𝑋 < 0) β†’ (βˆšβ€˜π‘‹) βˆ‰ ℝ)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47936
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