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Theorem List for Metamath Proof Explorer - 38001-38100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2reu4a 38001* Definition of double restricted existential uniqueness ("exactly one  x and exactly one  y"), analogous to 2eu4 2358 with the additional requirement that the restricting classes are not empty (which is not necessary as shown in 2reu4 38002). (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  (
 ( A  =/=  (/)  /\  B  =/= 
 (/) )  ->  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ph  ->  ( x  =  z  /\  y  =  w )
 ) ) ) )
 
Theorem2reu4 38002* Definition of double restricted existential uniqueness ("exactly one  x and exactly one  y"), analogous to 2eu4 2358. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ph  ->  ( x  =  z  /\  y  =  w )
 ) ) )
 
Theorem2reu7 38003* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2363. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  E! x  e.  A  E! y  e.  B  ( E. x  e.  A  ph 
 /\  E. y  e.  B  ph ) )
 
Theorem2reu8 38004* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2364. Curiously, we can put  E! on either of the internal conjuncts but not both. We can also commute  E! x  e.  A E! y  e.  B using 2reu7 38003. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( E! x  e.  A  E! y  e.  B  ( E. x  e.  A  ph 
 /\  E. y  e.  B  ph )  <->  E! x  e.  A  E! y  e.  B  ( E! x  e.  A  ph 
 /\  E. y  e.  B  ph ) )
 
21.33.2  Alternative definitions of function's and operation's values

The current definition of the value 
( F `  A
) of a function  F for an argument  A (see df-fv 5609) assures that this value is always a set, see fex 6153. 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 5905 and fvprc 5875).

Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from  ( F `  A
)  =  (/) alone it cannot be decided/derived if  ( F `  A ) is meaningful ( F is actually a function which is defined for  A and really has the function value  (/)) or not. Therefore, additional assumptions are required, such as  (/)  e/  ran  F,  (/)  e.  ran  F or 
Fun  F  /\  A  e. 
dom  F (see, for example, ndmfvrcl 5906).

To avoid such an ambiguity, an alternative definition  ( F''' A ) (see df-afv 38009) would be possible which evaluates to the universal class ( ( F''' A )  =  _V) if it is not meaningful (see afvnfundmuv 38031, ndmafv 38032, afvprc 38036 and nfunsnafv 38034), and which corresponds to the current definition ( ( F `  A )  =  ( F''' A )) if it is (see afvfundmfveq 38030). That means  ( F''' A )  =  _V  ->  ( F `  A )  =  (/) (see afvpcfv0 38038), but  ( F `  A )  =  (/)  ->  ( F''' A )  =  _V is not generally valid.

In the theory of partial functions, it is a common case that  F is not defined at  A, which also would result in  ( F''' A )  =  _V. In this context we say  ( F''' A ) "is not defined" instead of "is not meaningful".

With this definition the following intuitive equivalence holds:  ( F''' A )  e.  _V <-> " ( F''' A ) is meaningful/defined".

An interesting question would be if 
( F `  A
) could be replaced by  ( F''' A ) in most of the theorems based on function's values. If we look at the (currently 19) proofs using the definition df-fv 5609 of 
( F `  A
), we see that analogons for the following 8 theorems can be proven using the alternative definition: fveq1 5880-> afveq1 38026, fveq2 5881-> afveq2 38027, nffv 5888-> nfafv 38028, csbfv12 5916-> csbafv12g , fvres 5895-> afvres 38064, rlimdm 13593-> rlimdmafv 38069, tz6.12-1 5897-> tz6.12-1-afv 38066, fveu 5873-> afveu 38045.

3 theorems proved by directly using df-fv 5609 are within a mathbox (fvsb 36442) or not used (isumclim3 13798, avril1 25745).

However, the remaining 8 theorems proved by directly using df-fv 5609 are used more or less often:

* fvex 5891: used in about 1750 proofs.

* tz6.12-1 5897: root theorem of many theorems which have not a strict analogon, and which are used many times: fvprc 5875 (used in about 127 proofs), tz6.12i 5901 (used - indirectly via fvbr0 5902 and fvrn0 5903- in 18 proofs, and in fvclss 6162 used in fvclex 6779 used in fvresex 6780, which is not used!), dcomex 8875 (used in 4 proofs), ndmfv 5905 (used in 86 proofs) and nfunsn 5912 (used by dffv2 5954 which is not used).

* fv2 5876: only used by elfv 5879, which is only used by fv3 5894, which is not used.

* dffv3 5877: used by dffv4 5878 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTOLD 36853, csbfv12gALTVD 36936), by shftval 13116 (itself used in 9 proofs), by dffv5 30476 (mathbox) and by fvco2 5956, which has the analogon afvco2 38068.

* fvopab5 5989: used only by ajval 26348 (not used) and by adjval 27378 ( used - indirectly - in 9 proofs).

* zsum 13762: used (via isum 13763, sum0 13765 and fsumsers 13772) in more than 90 proofs.

* isumshft 13875: used in pserdv2 23250 and (via logtayl 23470) 4 other proofs.

* ovtpos 6996: used in 14 proofs.

As a result of this analysis we can say that the current definition of a function's value is crucial for Metamath and cannot be exchanged easily with an alternative definition. While fv2 5876, dffv3 5877, fvopab5 5989, zsum 13762, isumshft 13875 and ovtpos 6996 are not critical or are, hopefully, also valid for the alternative definition, fvex 5891 and tz6.12-1 5897 (and the theorems based on them) are essential for the current definition of function values.

With the same arguments, an alternatvie definition of operation's values (( A O B)) could be meaningful to avoid ambiguities, see df-aov 38010.

For additional discussions/material see https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4.

 
Syntaxwdfat 38005 Extend the definition of a wff to include the "defined at" predicate. (Read: (The Function)  F is defined at (the argument)  A). In a previous version, the token "def@" was used. However, since the @ is used (informally) as a replacement for $ in commented out sections that may be deleted some day. While there is no violation of any standard to use the @ in a token, it could make the search for such commented-out sections slightly more difficult. (See remark of Norman Megill at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
 wff  F defAt  A
 
Syntaxcafv 38006 Extend the definition of a class to include the value of a function. (Read: The value of  F at  A, or " F of  A."). In a previous version, the symbol " ' " was used. However, since the similarity with the symbol 
` used for the current definition of a function's value (see df-fv 5609), which, by the way, was intended to visualize that in many cases  ` and " ' " are exchangeable, makes reading the theorems, especially those which uses both definitions as dfafv2 38024, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 5609 and df-ima 4867. And not three backticks ( three times  ` ) since that would be annoying to escape in a comment. (See remark of Norman Megill and Gerard Lang at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
 class  ( F''' A )
 
Syntaxcaov 38007 Extend class notation to include the value of an operation  F (such as  +) for two arguments  A and  B. Note that the syntax is simply three class symbols in a row surrounded by a pair of parentheses in contrast to the current definition, see df-ov 6308.
 class (( A F B))
 
Definitiondf-dfat 38008 Definition of the predicate that determines if some class  F is defined as function for an argument  A or, in other words, if the function value for some class  F for an argument  A is defined. We say that  F is defined at  A if a  F is a function restricted to the member  A of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F defAt  A  <->  ( A  e.  dom 
 F  /\  Fun  ( F  |`  { A } )
 ) )
 
Definitiondf-afv 38009* Alternative definition of the value of a function,  ( F''' A ), also known as function application. In contrast to  ( F `  A )  =  (/) (see df-fv 5609 and ndmfv 5905),  ( F''' A )  =  _V if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F''' A )  =  if ( F defAt  A ,  ( iota x A F x ) ,  _V )
 
Definitiondf-aov 38010 Define the value of an operation. In contrast to df-ov 6308, the alternative definition for a function value (see df-afv 38009) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation  F and its arguments  A and  B- will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |- (( A F B))  =  ( F'''
 <. A ,  B >. )
 
21.33.2.1  Restricted quantification (extension)
 
Theoremralbinrald 38011* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
 |-  ( ph  ->  X  e.  A )   &    |-  ( x  e.  A  ->  x  =  X )   &    |-  ( x  =  X  ->  ( ps  <->  th ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 th ) )
 
21.33.2.2  The universal class (extension)
 
Theoremnvelim 38012 If a class is the universal class it doesn't belong to any class, generalisation of nvel 4564. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( A  =  _V  ->  -.  A  e.  B )
 
21.33.2.3  Introduce the Axiom of Power Sets (extension)
 
Theoremalneu 38013 If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.)
 |-  ( A. x ph  ->  -.  E! x ph )
 
Theoremeu2ndop1stv 38014* If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  ( E! y <. A ,  y >.  e.  V  ->  A  e.  _V )
 
21.33.2.4  Relations (extension)
 
Theoremeldmressn 38015 Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( B  e.  dom  ( F  |`  { A } )  ->  B  =  A )
 
21.33.2.5  Functions (extension)
 
Theoremfveqvfvv 38016 If a function's value at an argument is the universal class (which can never be the case because of fvex 5891), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything (see pm2.21i 134). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( F `  A )  =  _V  ->  ( F `  A )  =  B )
 
Theoremfunresfunco 38017 Composition of two functions, generalization of funco 5639. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( Fun  ( F  |` 
 ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
 
Theoremfnresfnco 38018 Composition of two functions, similar to fnco 5702. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( F  |`  ran  G )  Fn  ran  G  /\  G  Fn  B )  ->  ( F  o.  G )  Fn  B )
 
Theoremfuncoressn 38019 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( ( G `
  X )  e. 
 dom  F  /\  Fun  ( F  |`  { ( G `
  X ) }
 ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( ( F  o.  G )  |`  { X } ) )
 
Theoremfunressnfv 38020 A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( X  e.  dom  ( F  o.  G )  /\  Fun  ( ( F  o.  G )  |`  { X } ) ) 
 /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( F  |`  { ( G `
  X ) }
 ) )
 
21.33.2.6  Predicate "defined at"
 
Theoremdfateq12d 38021 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
 
Theoremnfdfat 38022 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g. for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |- 
 F/ x  F defAt  A
 
Theoremdfdfat2 38023* Alternate definition of the predicate "defined at" not using the  Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F defAt  A  <->  ( A  e.  dom 
 F  /\  E! y  A F y ) )
 
21.33.2.7  Alternative definition of the value of a function
 
Theoremdfafv2 38024 Alternative definition of  ( F''' A ) using  ( F `
 A ) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F''' A )  =  if ( F defAt  A ,  ( F `  A ) ,  _V )
 
Theoremafveq12d 38025 Equality deduction for function value, analogous to fveq12d 5887. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )
 
Theoremafveq1 38026 Equality theorem for function value, analogous to fveq1 5880. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F  =  G  ->  ( F''' A )  =  ( G''' A ) )
 
Theoremafveq2 38027 Equality theorem for function value, analogous to fveq1 5880. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  =  B  ->  ( F''' A )  =  ( F''' B ) )
 
Theoremnfafv 38028 Bound-variable hypothesis builder for function value, analogous to nffv 5888. To prove a deduction version of this analogous to nffvd 5890 is not easily possible because a deduction version of nfdfat 38022 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ x ( F''' A )
 
Theoremcsbafv12g 38029 Move class substitution in and out of a function value, analogous to csbfv12 5916, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6339. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ ( F''' B )  =  (
 [_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
 
Theoremafvfundmfveq 38030 If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F defAt  A  ->  ( F''' A )  =  ( F `
  A ) )
 
Theoremafvnfundmuv 38031 If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  F defAt  A  ->  ( F''' A )  =  _V )
 
Theoremndmafv 38032 The value of a class outside its domain is the universe, compare with ndmfv 5905. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
 
Theoremafvvdm 38033 If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F''' A )  e.  B  ->  A  e.  dom  F )
 
Theoremnfunsnafv 38034 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5912 (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  Fun  ( F  |`  { A } )  ->  ( F''' A )  =  _V )
 
Theoremafvvfunressn 38035 If the function 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, 25-May-2017.)
 |-  (
 ( F''' A )  e.  B  ->  Fun  ( F  |`  { A } ) )
 
Theoremafvprc 38036 A function's value at a proper class is the universe, compare with fvprc 5875. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  A  e.  _V  ->  ( F''' A )  =  _V )
 
Theoremafvvv 38037 If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F''' A )  e.  B  ->  A  e.  _V )
 
Theoremafvpcfv0 38038 If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F''' A )  =  _V  ->  ( F `  A )  =  (/) )
 
Theoremafvnufveq 38039 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
 
Theoremafvvfveq 38040 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
 
Theoremafv0fv0 38041 If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F''' A )  =  (/)  ->  ( F `  A )  =  (/) )
 
Theoremafvfvn0fveq 38042 If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F `  A )  =/=  (/)  ->  ( F''' A )  =  ( F `
  A ) )
 
Theoremafv0nbfvbi 38043 The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( (/)  e/  B  ->  ( ( F''' A )  e.  B  <->  ( F `  A )  e.  B ) )
 
Theoremafvfv0bi 38044 The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F `  A )  =  (/)  <->  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V ) )
 
Theoremafveu 38045* The value of a function at a unique point, analogous to fveu 5873. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
 )
 
Theoremfnbrafvb 38046 Equivalence of function value and binary relation, analogous to fnbrfvb 5921. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( ( F''' B )  =  C  <->  B F C ) )
 
Theoremfnopafvb 38047 Equivalence of function value and ordered pair membership, analogous to fnopfvb 5922. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( ( F''' B )  =  C  <->  <. B ,  C >.  e.  F ) )
 
Theoremfunbrafvb 38048 Equivalence of function value and binary relation, analogous to funbrfvb 5923. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( ( F''' A )  =  B  <->  A F B ) )
 
Theoremfunopafvb 38049 Equivalence of function value and ordered pair membership, analogous to funopfvb 5924. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( ( F''' A )  =  B  <->  <. A ,  B >.  e.  F ) )
 
Theoremfunbrafv 38050 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 5919. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) )
 
Theoremfunbrafv2b 38051 Function value in terms of a binary relation, analogous to funbrfv2b 5925. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fun  F  ->  ( A F B  <->  ( A  e.  dom 
 F  /\  ( F''' A )  =  B ) ) )
 
Theoremdfafn5a 38052* Representation of a function in terms of its values, analogous to dffn5 5926 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
 
Theoremdfafn5b 38053* Representation of a function in terms of its values, analogous to dffn5 5926 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( A. x  e.  A  ( F''' x )  e.  V  ->  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F''' x ) ) ) )
 
Theoremfnrnafv 38054* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 5927. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F''' x ) } )
 
Theoremafvelrnb 38055* A member of a function's range is a value of the function, analogous to fvelrnb 5928 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  V ) 
 ->  ( B  e.  ran  F  <->  E. x  e.  A  ( F''' x )  =  B ) )
 
Theoremafvelrnb0 38056* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 5928. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
 |-  ( F  Fn  A  ->  ( B  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  B ) )
 
Theoremdfaimafn 38057* Alternate definition of the image of a function, analogous to dfimafn 5930. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  C_  dom  F )  ->  ( F " A )  =  { y  |  E. x  e.  A  ( F''' x )  =  y } )
 
Theoremdfaimafn2 38058* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5931. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  C_  dom  F )  ->  ( F " A )  =  U_ x  e.  A  { ( F''' x ) } )
 
Theoremafvelima 38059* Function value in an image, analogous to fvelima 5933. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  ( F " B ) )  ->  E. x  e.  B  ( F''' x )  =  A )
 
Theoremafvelrn 38060 A function's value belongs to its range, analogous to fvelrn 6030. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( F''' A )  e.  ran  F )
 
Theoremfnafvelrn 38061 A function's value belongs to its range, analogous to fnfvelrn 6034. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( F''' B )  e.  ran  F )
 
Theoremfafvelrn 38062 A function's value belongs to its codomain, analogous to ffvelrn 6035. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B )
 
Theoremffnafv 38063* A function maps to a class to which all values belong, analogous to ffnfv 6064. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
 
Theoremafvres 38064 The value of a restricted function, analogous to fvres 5895. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  e.  B  ->  ( ( F  |`  B )''' A )  =  ( F''' A ) )
 
Theoremtz6.12-afv 38065* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 5898. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  (
 ( <. A ,  y >.  e.  F  /\  E! y <. A ,  y >.  e.  F )  ->  ( F''' A )  =  y )
 
Theoremtz6.12-1-afv 38066* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 5897. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  (
 ( A F y 
 /\  E! y  A F y )  ->  ( F''' A )  =  y
 )
 
Theoremdmfcoafv 38067 Domains of a function composition, analogous to dmfco 5955. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( Fun  G  /\  A  e.  dom  G ) 
 ->  ( A  e.  dom  ( F  o.  G ) 
 <->  ( G''' A )  e.  dom  F ) )
 
Theoremafvco2 38068 Value of a function composition, analogous to fvco2 5956. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( G  Fn  A  /\  X  e.  A ) 
 ->  ( ( F  o.  G )''' X )  =  ( F''' ( G''' X ) ) )
 
Theoremrlimdmafv 38069 Two ways to express that a function has a limit, analogous to rlimdm 13593. (Contributed by Alexander van der Vekens, 27-Nov-2017.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = +oo )   =>    |-  ( ph  ->  ( F  e.  dom  ~~> r  <->  F  ~~> r  (  ~~> r ''' F ) ) )
 
21.33.2.8  Alternative definition of the value of an operation
 
Theoremaoveq123d 38070 Equality deduction for operation value, analogous to oveq123d 6326. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  -> (( A F C))  = (( B G D))  )
 
Theoremnfaov 38071 Bound-variable hypothesis builder for operation value, analogous to nfov 6331. To prove a deduction version of this analogous to nfovd 6330 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 38028). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x (( A F B))
 
Theoremcsbaovg 38072 Move class substitution in and out of an operation. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( A  e.  D  ->  [_ A  /  x ]_ (( B F C))  = (( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C))  )
 
Theoremaovfundmoveq 38073 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.)
 |-  ( F defAt  <. A ,  B >.  -> (( A F B))  =  ( A F B ) )
 
Theoremaovnfundmuv 38074 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.)
 |-  ( -.  F defAt  <. A ,  B >.  -> (( A F B))  =  _V )
 
Theoremndmaov 38075 The value of an operation outside its domain, analogous to ndmafv 38032. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  <. A ,  B >.  e.  dom  F  -> (( A F B))  =  _V )
 
Theoremndmaovg 38076 The value of an operation outside its domain, analogous to ndmovg 6466. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( dom  F  =  ( R  X.  S ) 
 /\  -.  ( A  e.  R  /\  B  e.  S ) )  -> (( A F B))  =  _V )
 
Theoremaovvdm 38077 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.)
 |-  ( (( A F B))  e.  C  -> 
 <. A ,  B >.  e. 
 dom  F )
 
Theoremnfunsnaov 38078 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.)
 |-  ( -.  Fun  ( F  |`  { <. A ,  B >. } )  -> (( A F B))  =  _V )
 
Theoremaovvfunressn 38079 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.)
 |-  ( (( A F B))  e.  C  ->  Fun  ( F  |`  { <. A ,  B >. } )
 )
 
Theoremaovprc 38080 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 6335. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> (( A F B))  =  _V )
 
Theoremaovrcl 38081 Reverse closure for an operation value, analogous to afvvv 38037. In contrast to ovrcl 6338, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( (( A F B))  e.  C  ->  ( A  e.  _V  /\  B  e.  _V ) )
 
Theoremaovpcov0 38082 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.)
 |-  ( (( A F B))  =  _V  ->  ( A F B )  =  (/) )
 
Theoremaovnuoveq 38083 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.)
 |-  ( (( A F B))  =/=  _V  -> (( A F B))  =  ( A F B ) )
 
Theoremaovvoveq 38084 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.)
 |-  ( (( A F B))  e.  C  -> (( A F B))  =  ( A F B ) )
 
Theoremaov0ov0 38085 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.)
 |-  ( (( A F B))  =  (/)  ->  ( A F B )  =  (/) )
 
Theoremaovovn0oveq 38086 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.)
 |-  (
 ( A F B )  =/=  (/)  -> (( A F B))  =  ( A F B ) )
 
Theoremaov0nbovbi 38087 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.)
 |-  ( (/)  e/  C  ->  ( (( A F B))  e.  C  <->  ( A F B )  e.  C ) )
 
Theoremaovov0bi 38088 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.)
 |-  (
 ( A F B )  =  (/)  <->  ( (( A F B))  =  (/)  \/ (( A F B))  =  _V ) )
 
Theoremrspceaov 38089* A frequently used special case of rspc2ev 3199 for operation values, analogous to rspceov 6344. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( C  e.  A  /\  D  e.  B  /\  S  = (( C F D))  )  ->  E. x  e.  A  E. y  e.  B  S  = (( x F y))  )
 
Theoremfnotaovb 38090 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5922. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( F  Fn  ( A  X.  B )  /\  C  e.  A  /\  D  e.  B )  ->  ( (( C F D))  =  R  <->  <. C ,  D ,  R >.  e.  F ) )
 
Theoremffnaov 38091* An operation maps to a class to which all values belong, analogous to ffnov 6414. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B ) 
 /\  A. x  e.  A  A. y  e.  B (( x F y))  e.  C ) )
 
Theoremfaovcl 38092 Closure law for an operation, analogous to fovcl 6415. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F : ( R  X.  S ) --> C   =>    |-  ( ( A  e.  R  /\  B  e.  S )  -> (( A F B))  e.  C )
 
Theoremaovmpt4g 38093* Value of a function given by the "maps to" notation, analogous to ovmpt4g 6433. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  -> (( x F y))  =  C )
 
Theoremaoprssdm 38094* Domain of closure of an operation. In contrast to oprssdm 6464, no additional property for S (
-.  (/)  e.  S) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( x  e.  S  /\  y  e.  S )  -> (( x F y))  e.  S )   =>    |-  ( S  X.  S )  C_  dom  F
 
Theoremndmaovcl 38095 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6468 where it is required that the domain contains the empty set ( (/) 
e.  S). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   &    |-  ( ( A  e.  S  /\  B  e.  S )  -> (( A F B))  e.  S )   &    |- (( A F B))  e.  _V   =>    |- (( A F B))  e.  S
 
Theoremndmaovrcl 38096 Reverse closure law, in contrast to ndmovrcl 6469 where it is required that the operation's domain doesn't contain the empty set ( -.  (/)  e.  S), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( (( A F B))  e.  S  ->  ( A  e.  S  /\  B  e.  S ) )
 
Theoremndmaovcom 38097 Any operation is commutative outside its domain, analogous to ndmovcom 6470. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S )  -> (( A F B))  = (( B F A))  )
 
Theoremndmaovass 38098 Any operation is associative outside its domain. In contrast to ndmovass 6471 where it is required that the operation's domain doesn't contain the empty set ( -.  (/)  e.  S), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  -> (( (( A F B))  F C))  = (( A F (( B F C)) ))  )
 
Theoremndmaovdistr 38099 Any operation is distributive outside its domain. In contrast to ndmovdistr 6472 where it is required that the operation's domain doesn't contain the empty set (
-.  (/)  e.  S), no additional assumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  dom  F  =  ( S  X.  S )   &    |-  dom  G  =  ( S  X.  S )   =>    |-  ( -.  ( A  e.  S  /\  B  e.  S  /\  C  e.  S ) 
 -> (( A G (( B F C)) ))  = (( (( A G B))  F (( A G C)) ))  )
 
21.33.3  General auxiliary theorems
 
21.33.3.1  Miscellanea
 
Theorem1t10e1p1e11 38100 11 is 1 times 10 to the power of 1, plus 1. (Contributed by AV, 4-Aug-2020.)
 |- ; 1 1  =  ( ( 1  x.  ( 10 ^ 1 ) )  +  1 )
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