HomeHome Metamath Proof Explorer
Theorem List (p. 281 of 322)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-32154)
 

Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremafvfundmfveq 28001 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 28002 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 28003 The value of a class outside its domain is the universe, compare with ndmfv 5552. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
 
Theoremafvvdm 28004 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 28005 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5558 (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  Fun  ( F  |`  { A } )  ->  ( F''' A )  =  _V )
 
Theoremafvvfunressn 28006 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 28007 A function's value at a proper class is the universe, compare with fvprc 5519. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  A  e.  _V  ->  ( F''' A )  =  _V )
 
Theoremafvvv 28008 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 28009 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 28010 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 28011 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 28012 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 28013 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 28014 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 28015 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 ) )
 
Theoremfnbrafvb 28016 Equivalence of function value and binary relation, analogous to fnbrfvb 5563. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( ( F''' B )  =  C  <->  B F C ) )
 
Theoremfnopafvb 28017 Equivalence of function value and ordered pair membership, analogous to fnopfvb 5564. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( ( F''' B )  =  C  <->  <. B ,  C >.  e.  F ) )
 
Theoremfunbrafvb 28018 Equivalence of function value and binary relation, analogous to funbrfvb 5565. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( ( F''' A )  =  B  <->  A F B ) )
 
Theoremfunopafvb 28019 Equivalence of function value and ordered pair membership, analogous to funopfvb 5566. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( ( F''' A )  =  B  <->  <. A ,  B >.  e.  F ) )
 
Theoremfunbrafv 28020 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 5561. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) )
 
Theoremfunbrafv2b 28021 Function value in terms of a binary relation, analogous to funbrfv2b 5567. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fun  F  ->  ( A F B  <->  ( A  e.  dom 
 F  /\  ( F''' A )  =  B ) ) )
 
Theoremdfafn5a 28022* Representation of a function in terms of its values, analogous to dffn5 5568 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
 
Theoremdfafn5b 28023* Representation of a function in terms of its values, analogous to dffn5 5568 (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 28024* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 5569. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F''' x ) } )
 
Theoremafvelrnb 28025* A member of a function's range is a value of the function, analogous to fvelrnb 5570 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 28026* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 5570. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
 |-  ( F  Fn  A  ->  ( B  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  B ) )
 
Theoremdfaimafn 28027* Alternate definition of the image of a function, analogous to dfimafn 5571. (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 28028* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5572. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  C_  dom  F )  ->  ( F " A )  =  U_ x  e.  A  { ( F''' x ) } )
 
Theoremafvelima 28029* Function value in an image, analogous to fvelima 5574. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  ( F " B ) )  ->  E. x  e.  B  ( F''' x )  =  A )
 
Theoremafvelrn 28030 A function's value belongs to its range, analogous to fvelrn 5661. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( F''' A )  e.  ran  F )
 
Theoremfnafvelrn 28031 A function's value belongs to its range, analogous to fnfvelrn 5662. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( F''' B )  e.  ran  F )
 
Theoremfafvelrn 28032 A function's value belongs to its codomain, analogous to ffvelrn 5663. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B )
 
Theoremffnafv 28033* A function maps to a class to which all values belong, analogous to ffnfv 5685. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
 
Theoremafvres 28034 The value of a restricted function, analogous to fvres 5542. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  e.  B  ->  ( ( F  |`  B )''' A )  =  ( F''' A ) )
 
Theoremtz6.12-afv 28035* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, , analogous to tz6.12-1 5544, but it is required for A to be a set. (Contributed by Alexander van der Vekens, 28-Jul-2017.)
 |-  ( A  e.  V  ->  ( ( A F y 
 /\  E! y  A F y )  ->  ( F''' A )  =  y
 ) )
 
Theoremdmfcoafv 28036 Domains of a function composition, analogous to dmfco 5593. (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 28037 Value of a function composition, analogous to fvco2 5594. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( G  Fn  A  /\  X  e.  A ) 
 ->  ( ( F  o.  G )''' X )  =  ( F''' ( G''' X ) ) )
 
18.23.2.7  Alternative definition of the value of an operation
 
Theoremaoveq123d 28038 Equality deduction for operation value, analogous to oveq123d 5879. (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 28039 Bound-variable hypothesis builder for operation value, analogous to nfov 5881. To prove a deduction version of this analogous to nfovd 5880 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 27999). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x (( A F B))
 
Theoremcsbaovg 28040 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 28041 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 28042 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 28043 The value of an operation outside its domain, analogous to ndmafv 28003. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  <. A ,  B >.  e.  dom  F  -> (( A F B))  =  _V )
 
Theoremndmaovg 28044 The value of an operation outside its domain, analogous to ndmovg 6003. (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 28045 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 28046 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 28047 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 28048 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 5885. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> (( A F B))  =  _V )
 
Theoremaovrcl 28049 Reverse closure for an operation value, analogous to afvvv 28008. In contrast to ovrcl 5888, 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 28050 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 28051 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 28052 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 28053 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 28054 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 28055 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 28056 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 28057* A frequently used special case of rspc2ev 2892 for operation values, analogous to rspceov 5893. (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 28058 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5564. (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 28059* An operation maps to a class to which all values belong, analogous to ffnov 5948. (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 28060 Closure law for an operation, analogous to fovcl 5949. (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 28061* Value of a function given by the "maps to" notation, analogous to ovmpt4g 5970. (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 28062* Domain of closure of an operation. In contrast to oprssdm 6002, 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 28063 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6005 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 28064 Reverse closure law, in contrast to ndmovrcl 6006 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 28065 Any operation is commutative outside its domain, analogous to ndmovcom 6007. (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 28066 Any operation is associative outside its domain. In contrast to ndmovass 6008 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 28067 Any operation is distributive outside its domain. In contrast to ndmovdistr 6009 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)) ))  )
 
18.23.3  Graph theory

Until now (7-Oct-2017), there are no definitions and theorems for Graph Theory in the main part of set.mm. However, the Mathbox of Mario Carneiro contains definitions for undirected multigraphs (df-umgra 23863), for the degree of a vertex (df-vdgr 23865) and for Eulerian Paths (df-eupa 23864). These definitions (and the corresponding theorems) are not used within this section, except for the proofs of the compatibility between the definitions (see uslisumgra 28112 and usisumgra 28114).

 
18.23.3.1  Unordered and ordered pairs (extension)
 
Theoremdifprsneq 28068 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A } )
 
Theoremdifprsng 28069 Removal of a singleton from a (proper) unordered pair. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( A  =/=  B  ->  ( { A ,  B }  \  { A } )  =  { B } )
 
Theoremdiftpsneq 28070 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  (
 ( A  =/=  C  /\  B  =/=  C ) 
 ->  ( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
 
Theoremtppreq3 28071 An unordered triple is an unordered pair if one of its elemets is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B }
 )
 
Theoremtpprceq3 28072 An unordered triple is an unordered pair if one of its elemets is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( -.  ( C  e.  _V  /\  C  =/=  B ) 
 ->  { A ,  B ,  C }  =  { A ,  B }
 )
 
Theoremprneimg 28073 Two pairs are not equal if one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  (
 ( ( A  e.  U  /\  B  e.  V )  /\  ( C  e.  X  /\  D  e.  Y ) )  ->  ( ( ( A  =/=  C  /\  A  =/=  D )  \/  ( B  =/=  C 
 /\  B  =/=  D ) )  ->  { A ,  B }  =/=  { C ,  D }
 ) )
 
Theoremprelpw 28074 A pair of elements of a set is an element of the set's power set. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  (
 ( A  e.  V  /\  B  e.  V ) 
 ->  ( P  =  { A ,  B }  ->  P  e.  ~P V ) )
 
18.23.3.2  Functions (extension)
 
Theoremf1oprg 28075 An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5515. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y ) )  ->  ( ( A  =/=  C  /\  B  =/=  D )  ->  { <. A ,  B >. ,  <. C ,  D >. } : { A ,  C } -1-1-onto-> { B ,  D } ) )
 
Theoremf1oun2prg 28076 A union of unordered pairs of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y ) )  ->  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D ) )  ->  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  D >. } ) : ( { 0 ,  1 }  u.  { 2 ,  3 } ) -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) ) )
 
18.23.3.3  Operations (Extension)
 
Theoremnssdmovg 28077 The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.)
 |-  (
 ( dom  F  C_  ( R  X.  S )  /\  -.  ( A  e.  R  /\  B  e.  S ) )  ->  ( A F B )  =  (/) )
 
Theoremmpt2ndm0 28078* The value of an operation given by a maps-to rule is the empty set if the arguments ar not contained in the base sets of the rule. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  F  =  ( x  e.  X ,  y  e.  Y  |->  C )   =>    |-  ( -.  ( V  e.  X  /\  W  e.  Y )  ->  ( V F W )  =  (/) )
 
18.23.3.4  "Maps to" notation (Extension)

The following theorems are about maps-to operations ( see df-mpt2 5863) where the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 5924, ovmpt2x 5976 and fmpt2x 6190). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short.

 
Theoremmpt2xopn0yelv 28079* If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  ( <. V ,  W >. F K )  ->  K  e.  V )
 )
 
Theoremmpt2xopynvov0g 28080* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e/  V )  ->  ( <. V ,  W >. F K )  =  (/) )
 
Theoremmpt2xopxnop0 28081* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( -.  V  e.  ( _V  X.  _V )  ->  ( V F K )  =  (/) )
 
Theoremmpt2xopx0ov0 28082* If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( (/) F K )  =  (/)
 
Theoremmpt2xopxprcov0 28083* If the components of the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( -.  ( V  e.  _V  /\  W  e.  _V )  ->  ( <. V ,  W >. F K )  =  (/) )
 
Theoremmpt2xopynvov0 28084* If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  C )   =>    |-  ( K  e/  V  ->  ( <. V ,  W >. F K )  =  (/) )
 
Theoremmpt2xopoveq 28085* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )   =>    |-  ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [.
 <. V ,  W >.  /  x ]. [. K  /  y ]. ph } )
 
Theoremmpt2xopovel 28086* Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )   =>    |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  ( <. V ,  W >. F K )  <->  ( K  e.  V  /\  N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
 ) )
 
Theoremmpt2xopoveqd 28087* Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )   &    |-  ( ps  ->  ( V  e.  X  /\  W  e.  Y )
 )   &    |-  ( ( ps  /\  -.  K  e.  V ) 
 ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  =  (/) )   =>    |-  ( ps  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [.
 <. V ,  W >.  /  x ]. [. K  /  y ]. ph } )
 
18.23.3.5  The ` # ` (finite set size) function (extension)
 
Theoremelprchashprn2 28088 If one element of an unordered pair is not a set, the size of the unordered pair is not 2. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  ( -.  M  e.  _V  ->  -.  ( # `  { M ,  N } )  =  2 )
 
18.23.3.6  Longer string literals (extension)
 
Theorems2prop 28089 A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( A  e.  S  /\  B  e.  S ) 
 ->  <" A B ">  =  { <. 0 ,  A >. ,  <. 1 ,  B >. } )
 
Theorems4prop 28090 A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B C D ">  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
 <. 3 ,  D >. } ) )
 
Theorems2f1o 28091 A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( A  e.  S  /\  B  e.  S  /\  A  =/=  B )  ->  ( E  =  <" A B ">  ->  E : { 0 ,  1 } -1-1-onto-> { A ,  B } ) )
 
Theorems4dom 28092 The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  (
 ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S ) )  ->  ( E  =  <" A B C D ">  ->  dom 
 E  =  ( {
 0 ,  1 }  u.  { 2 ,  3 } ) ) )
 
Theorems4f1o 28093 A length 4 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  (
 ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S ) )  ->  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D ) )  ->  ( E  =  <" A B C D ">  ->  E : dom  E -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) ) ) )
 
18.23.3.7  Undirected simple graphs

Although undirected simple graphs (with or without loops) are defined separately from undirected mulitigraphs (see df-umgra 23863), the definitions are similar and therefore compatible with each other, see uslisumgra 28112 and usisuslgra 28113.

 
Syntaxcuslg 28094 Extend class notation with undirected (simple) graphs with loops.
 class USLGrph
 
Syntaxcusg 28095 Extend class notation with undirected (simple) graphs (without loops).
 class USGrph
 
Definitiondf-uslgra 28096* Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph  <. V ,  E >. where  E is an injective (one-to-one) function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a multigraph, there is at most one edge between two vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Definitiondf-usgra 28097* Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph  <. V ,  E >. where 
E is an injective (one-to-one) function into subsets of  V of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itsself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
 
Theoremreluslgra 28098 The class of all undirected simple graph with loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  Rel USLGrph
 
Theoremrelusgra 28099 The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  Rel USGrph
 
Theoremuslgrav 28100 The classes of vertices and edges of an undirected simple graph with loops are sets. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USLGrph  E  ->  ( V  e.  _V  /\  E  e.  _V ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 32101-32154
  Copyright terms: Public domain < Previous  Next >