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Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfafvelrn 27901 A function's value belongs to its codomain, analogous to ffvelrn 5827. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B )
 
Theoremffnafv 27902* A function maps to a class to which all values belong, analogous to ffnfv 5853. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
 
Theoremafvres 27903 The value of a restricted function, analogous to fvres 5704. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  e.  B  ->  ( ( F  |`  B )''' A )  =  ( F''' A ) )
 
Theoremtz6.12-afv 27904* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 5707. (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 27905* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 5706. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  (
 ( A F y 
 /\  E! y  A F y )  ->  ( F''' A )  =  y
 )
 
Theoremdmfcoafv 27906 Domains of a function composition, analogous to dmfco 5756. (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 27907 Value of a function composition, analogous to fvco2 5757. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( G  Fn  A  /\  X  e.  A ) 
 ->  ( ( F  o.  G )''' X )  =  ( F''' ( G''' X ) ) )
 
Theoremrlimdmafv 27908 Two ways to express that a function has a limit, analogous to rlimdm 12300. (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 ) ) )
 
19.22.2.8  Alternative definition of the value of an operation
 
Theoremaoveq123d 27909 Equality deduction for operation value, analogous to oveq123d 6061. (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 27910 Bound-variable hypothesis builder for operation value, analogous to nfov 6063. To prove a deduction version of this analogous to nfovd 6062 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 27867). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x (( A F B))
 
Theoremcsbaovg 27911 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 27912 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 27913 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 27914 The value of an operation outside its domain, analogous to ndmafv 27871. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  <. A ,  B >.  e.  dom  F  -> (( A F B))  =  _V )
 
Theoremndmaovg 27915 The value of an operation outside its domain, analogous to ndmovg 6189. (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 27916 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 27917 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 27918 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 27919 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 6067. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> (( A F B))  =  _V )
 
Theoremaovrcl 27920 Reverse closure for an operation value, analogous to afvvv 27876. In contrast to ovrcl 6070, 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 27921 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 27922 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 27923 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 27924 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 27925 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 27926 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 27927 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 27928* A frequently used special case of rspc2ev 3020 for operation values, analogous to rspceov 6075. (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 27929 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5727. (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 27930* An operation maps to a class to which all values belong, analogous to ffnov 6133. (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 27931 Closure law for an operation, analogous to fovcl 6134. (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 27932* Value of a function given by the "maps to" notation, analogous to ovmpt4g 6155. (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 27933* Domain of closure of an operation. In contrast to oprssdm 6187, 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 27934 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6191 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 27935 Reverse closure law, in contrast to ndmovrcl 6192 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 27936 Any operation is commutative outside its domain, analogous to ndmovcom 6193. (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 27937 Any operation is associative outside its domain. In contrast to ndmovass 6194 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 27938 Any operation is distributive outside its domain. In contrast to ndmovdistr 6195 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)) ))  )
 
19.22.3  Auxiliary theorems for graph theory

Additional theorems for classical first order logic with equality, ZF set theory and theory of real and complex numbers used for proving the theorems for graph theory.

 
19.22.3.1  Negated equality and membership - extension
 
Theoremeqneqall 27939 A contradiction concerning equality implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  =  B  ->  ( A  =/=  B  ->  ph ) )
 
Theoremelnelall 27940 A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( A  e.  B  ->  ( A  e/  B  ->  ph ) )
 
19.22.3.2  "Weak deduction theorem" for set theory - extension
 
Theorem2if2 27941 Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
 |-  (
 ( ph  /\  ps )  ->  D  =  A )   &    |-  ( ( ph  /\  -.  ps 
 /\  th )  ->  D  =  B )   &    |-  ( ( ph  /\ 
 -.  ps  /\  -.  th )  ->  D  =  C )   =>    |-  ( ph  ->  D  =  if ( ps ,  A ,  if ( th ,  B ,  C ) ) )
 
19.22.3.3  Power classes - extension
 
Theorem3xpexg 27942 The cross product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
 |-  ( V  e.  W  ->  ( ( V  X.  V )  X.  V )  e. 
 _V )
 
19.22.3.4  Unordered and ordered pairs - extension
 
Theoremnelprd 27943 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  -.  A  e.  { B ,  C } )
 
Theorempr1eqbg 27944 A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =  C 
 <->  { A ,  B }  =  { B ,  C } ) )
 
Theorempr1nebg 27945 A (proper) pair is not equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X ) 
 /\  A  =/=  B )  ->  ( A  =/=  C  <->  { A ,  B }  =/=  { B ,  C } ) )
 
Theoremraldifsnb 27946* Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( A. x  e.  A  ( x  =/=  Y  ->  ph )  <->  A. x  e.  ( A  \  { Y }
 ) ph )
 
Theoremrexdifpr 27947 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
 |-  ( E. x  e.  ( A  \  { B ,  C } ) ph  <->  E. x  e.  A  ( x  =/=  B  /\  x  =/=  C  /\  ph )
 )
 
Theoremel2xptp 27948* A member of a nested cross product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( A  e.  ( ( B  X.  C )  X.  D )  <->  E. x  e.  B  E. y  e.  C  E. z  e.  D  A  =  <. x ,  y ,  z >. )
 
Theoremel2xptp0 27949 A member of a nested cross product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  (
 ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( ( A  e.  ( ( U  X.  V )  X.  W ) 
 /\  ( ( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) )  <->  A  =  <. X ,  Y ,  Z >. ) )
 
Theoremotel3xp 27950 An ordered triple is an element of a doubled cross product. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
 |-  (
 ( T  =  <. A ,  B ,  C >.  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) )  ->  T  e.  ( ( X  X.  Y )  X.  Z ) )
 
Theoremoteqimp 27951 The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 ->  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  B  /\  ( 2nd `  T )  =  C )
 ) )
 
Theoremotthg 27952 Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  (
 ( A  e.  U  /\  B  e.  V  /\  C  e.  W )  ->  ( <. A ,  B ,  C >.  =  <. D ,  E ,  F >.  <-> 
 ( A  =  D  /\  B  =  E  /\  C  =  F )
 ) )
 
Theoremotsndisj 27953* The singletons consisting of ordered triples which have distinct third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  (
 ( A  e.  X  /\  B  e.  Y ) 
 -> Disj  c  e.  V { <. A ,  B ,  c >. } )
 
Theoremotiunsndisj 27954* 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.)
 |-  ( B  e.  X  -> Disj  a  e.  V U_ c  e.  ( W  \  {
 a } ) { <. a ,  B ,  c >. } )
 
TheoremotiunsndisjX 27955* 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.)
 |-  ( B  e.  X  -> Disj  a  e.  V U_ c  e.  W  { <. a ,  B ,  c >. } )
 
19.22.3.5  Indexed union and intersection - extension
 
Theoremiunxprg 27956* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A ,  B } C  =  ( D  u.  E ) )
 
19.22.3.6  Relations - extension
 
Theoremresisresindm 27957 The restriction of a relation by a set  B is identical with the restriction by the intersection of  B with the domain of the relation. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  ( Rel  F  ->  ( F  |`  B )  =  ( F  |`  ( B  i^i  dom  F ) ) )
 
19.22.3.7  Functions - extension
 
Theorem2f1fvneq 27958 If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  (
 ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C )  /\  A  =/=  B ) 
 ->  ( ( ( E `
  ( F `  A ) )  =  X  /\  ( E `
  ( F `  B ) )  =  Y )  ->  X  =/=  Y ) )
 
Theoremdff14a 27959* A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  ( F `  x )  =/=  ( F `  y ) ) ) )
 
Theoremdff14b 27960* A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  ( A  \  { x } ) ( F `
  x )  =/=  ( F `  y
 ) ) )
 
Theoremf12dfv 27961 A one-to-one function with a pair as domain in terms of function values. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  A  =  { X ,  Y }   =>    |-  ( ( ( X  e.  U  /\  Y  e.  V )  /\  X  =/=  Y )  ->  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  ( F `
  X )  =/=  ( F `  Y ) ) ) )
 
Theoremf13dfv 27962 A one-to-one function with a triple as domain in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  A  =  { X ,  Y ,  Z }   =>    |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  /\  ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z ) )  ->  ( F : A -1-1-> B  <->  ( F : A
 --> B  /\  ( ( F `  X )  =/=  ( F `  Y )  /\  ( F `
  X )  =/=  ( F `  Z )  /\  ( F `  Y )  =/=  ( F `  Z ) ) ) ) )
 
Theoremf13idfv 27963 A one-to-one function with a set of 3 sequential integers as domain in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  A  =  ( 0 ... 2
 )   =>    |-  ( F : A -1-1-> B  <-> 
 ( F : A --> B  /\  ( ( F `
  0 )  =/=  ( F `  1
 )  /\  ( F `  0 )  =/=  ( F `  2 )  /\  ( F `  1 )  =/=  ( F `  2 ) ) ) )
 
Theoremrnfdmpr 27964 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  e.  V  /\  Y  e.  W ) 
 ->  ( F  Fn  { X ,  Y }  ->  ran  F  =  {
 ( F `  X ) ,  ( F `  Y ) } )
 )
 
Theoremimarnf1pr 27965 The image of the range of a function  F under a function  E when  F is a function of a pair into the domain of  E. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  (
 ( X  e.  V  /\  Y  e.  W ) 
 ->  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R ) 
 /\  ( ( E `
  ( F `  X ) )  =  A  /\  ( E `
  ( F `  Y ) )  =  B ) )  ->  ( E " ran  F )  =  { A ,  B } ) )
 
19.22.3.8  Equinumerosity - extension
 
Theoremresfnfinfin 27966 The restriction of a function by a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  (
 ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  B )  e.  Fin )
 
Theorem3xpfi 27967 The cross product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( V  e.  Fin  ->  (
 ( V  X.  V )  X.  V )  e. 
 Fin )
 
19.22.3.9  Subtraction - extension
 
Theoremcnm1cn 27968 A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  ( N  e.  CC  ->  ( N  -  1 )  e.  CC )
 
19.22.3.10  Multiplication - extension
 
Theoremkcnktkm1cn 27969 k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
 
19.22.3.11  Ordering on reals (cont.) - extension
 
Theoremleaddsuble 27970 Addition and subtraction on one side of 'less or equal'. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  ( ( A  +  B )  -  C )  <_  A ) )
 
19.22.3.12  Nonnegative integers (as a subset of complex numbers) - extension
 
Theorem0mnnnnn0 27971 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
 |-  ( N  e.  NN  ->  ( 0  -  N ) 
 e/  NN0 )
 
Theoremlesubnn0 27972 Subtracting a nonnegative integer from a nonnegative integer which is greater than or equal to the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( B  <_  A  ->  ( A  -  B )  e.  NN0 ) )
 
Theoremltsubnn0 27973 Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( B  <  A  ->  ( A  -  B )  e.  NN0 ) )
 
Theoremnn0readdcl 27974 Closure law for addition of reals, restricted to nonnnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  +  B )  e.  RR )
 
Theoremnn0resubcl 27975 Closure law for subtraction of reals, restricted to nonnnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  -  B )  e.  RR )
 
19.22.3.13  Finite intervals of integers - extension
 
Theoremssfz12 27976 Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( K  e.  ZZ  /\  L  e.  ZZ  /\  K  <_  L )  ->  ( ( K ... L )  C_  ( M ... N )  ->  ( M  <_  K  /\  L  <_  N ) ) )
 
Theoremelfzmlbm 27977 Subtracting the left border of a finite sets of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
 |-  ( K  e.  ( M ... N )  ->  ( K  -  M )  e.  ( 0 ... ( N  -  M ) ) )
 
Theoremelfzmlbp 27978 Subtracting the lower bound of a finite sets of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
 |-  (
 ( N  e.  ZZ  /\  K  e.  ( M
 ... ( M  +  N ) ) ) 
 ->  ( K  -  M )  e.  ( 0 ... N ) )
 
Theoremnn0fz0 27979 Nonnegative integers expressed in terms of finite sets of sequential integers with lower bound 0. (Contributed by Alexander van der Vekens, 2-Apr-2018.)
 |-  ( N  e.  NN0  <->  N  e.  (
 0 ... N ) )
 
Theoremzletr 27980 Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
 |-  (
 ( J  e.  ZZ  /\  K  e.  ZZ  /\  L  e.  ZZ )  ->  ( ( J  <_  K 
 /\  K  <_  L )  ->  J  <_  L ) )
 
Theoremelfzelfzelfz 27981 En element of a finite set of sequential integers is an element of a finite set of sequential integers with the upper bound being an element of the finite set of sequential integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
 |-  (
 ( K  e.  (
 0 ... N )  /\  L  e.  ( K ... N ) )  ->  K  e.  ( 0 ... L ) )
 
Theoremelfzelfzadd 27982 An element of a finite set of sequential integers is an element of an extended finite set of sequential integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( N  e.  (
 0 ... A )  ->  N  e.  ( 0 ... ( A  +  B ) ) ) )
 
Theorem0elfz 27983 0 is an element of a finite set of sequential integers from 0 to a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  ( N  e.  NN0  ->  0  e.  ( 0 ... N ) )
 
Theorem2elfz3nn0 27984 If there are two elements in a finite set of sequential integers from 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( A  e.  NN0  /\  B  e.  NN0  /\  N  e.  NN0 ) )
 
19.22.3.14  Half-open integer ranges (extension)
 
Theoremfzo0ss1 27985 Subset relationship for half-open sequences of integers with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 1..^ N )  C_  ( 0..^ N )
 
Theoremubmelfzo 27986 If an integer between 0 and an upper bound of a half open interval of integers is subtracted from this upper bound, the result is contained in this half open interval. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  ( K  e.  ( 1..^ N )  ->  ( N  -  K )  e.  ( 0..^ N ) )
 
Theoremubmelm1fzo 27987 If an integer between 0 and an upper bound of a half open interval of integers minus 1 is subtracted from this upper bound, the result is contained in this half open interval. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  ( K  e.  ( 1..^ N )  ->  ( ( N  -  K )  -  1 )  e.  ( 0..^ N ) )
 
Theoremfzo1fzo0n0 27988 An integer between 1 and an upper bound of a half open interval of integers is not 0 and between 0 and the upper bound of a half open interval of integers. (Contributed by Alexander van der Vekens, 21-Mar-2018.)
 |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( 0..^ N )  /\  K  =/=  0 ) )
 
Theoremelfzomelpfzo 27989 An integer increased by another interger is an element of a half-open range of integers if and only if the integer is contained in the half-open range of integers with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  (
 ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  L  e.  ZZ ) )  ->  ( K  e.  ( ( M  -  L )..^ ( N  -  L ) )  <->  ( K  +  L )  e.  ( M..^ N ) ) )
 
Theoremssfzo12 27990 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( K  e.  ZZ  /\  L  e.  ZZ  /\  K  <  L )  ->  ( ( K..^ L )  C_  ( M..^ N )  ->  ( M  <_  K 
 /\  L  <_  N ) ) )
 
Theoremfzosplitsnm1 27991 Removing a singleton from a half-open range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ( ZZ>= `  ( A  +  1
 ) ) )  ->  ( A..^ B )  =  ( ( A..^ ( B  -  1 ) )  u.  { ( B  -  1 ) }
 ) )
 
Theoremelfzonelfzo 27992 If an element of a half-open range of integers is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  ( N  e.  ZZ  ->  ( ( K  e.  ( M..^ R )  /\  -.  K  e.  ( M..^ N ) )  ->  K  e.  ( N..^ R ) ) )
 
Theoremfseq0hash 27993 The value of the size function on a finite 0-based sequence. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
 |-  (
 ( N  e.  NN0  /\  F  Fn  ( 0..^ N ) )  ->  ( # `  F )  =  N )
 
19.22.3.15  The ` # ` (finite set size) function - extension
 
Theoremhashimarn 27994 The size of the image of a one-to-one function  E under the range of a function  F which is a one-to-one function into the domain of  E equals the size of the function  F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
 |-  (
 ( E : dom  E
 -1-1-> ran  E  /\  E  e.  V )  ->  ( F : ( 0..^ ( # `  F ) )
 -1-1-> dom  E  ->  ( # `
  ( E " ran  F ) )  =  ( # `  F ) ) )
 
Theoremhashimarni 27995 If the size of the image of a one-to-one function  E under the range of a function  F which is a one-to-one function into the domain of  E is a nonnegative integer, the size of the function  F is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
 |-  (
 ( E : dom  E
 -1-1-> ran  E  /\  E  e.  V )  ->  (
 ( F : ( 0..^ ( # `  F ) ) -1-1-> dom  E  /\  P  =  ( E
 " ran  F )  /\  ( # `  P )  =  N )  ->  ( # `  F )  =  N )
 )
 
Theoremhashfirdm 27996 The size of a function with a half-open range of integers, starting with 0, as domain equals the right border of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  (
 ( N  e.  NN0  /\  F : ( 0..^ N ) --> B ) 
 ->  ( # `  F )  =  N )
 
Theoremhashfzdm 27997 The size of a function with a finite set of sequential integers, starting with 0, as domain equals the right border of this range increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
 |-  (
 ( N  e.  NN0  /\  F : ( 0
 ... N ) --> B ) 
 ->  ( # `  F )  =  ( N  +  1 ) )
 
Theoremeuhash1 27998* The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018.)
 |-  ( V  e.  W  ->  ( ( # `  V )  =  1  <->  E! a  a  e.  V ) )
 
19.22.3.16  Words over a set - extension
 
Theoremiswrd0i 27999 A zero-based sequence is a word. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( W : ( 0 ...
 L ) --> S  ->  W  e. Word  S )
 
Theoremswrdltnd 28000 The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( W  e. Word  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( L  <_  F  ->  ( W substr  <. F ,  L >. )  =  (/) ) )
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