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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremafveq12d 28101 Equality deduction for function value, analogous to fveq12d 5547. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )
 
Theoremafveq1 28102 Equality theorem for function value, analogous to fveq1 5540. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F  =  G  ->  ( F''' A )  =  ( G''' A ) )
 
Theoremafveq2 28103 Equality theorem for function value, analogous to fveq1 5540. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  =  B  ->  ( F''' A )  =  ( F''' B ) )
 
Theoremnfafv 28104 Bound-variable hypothesis builder for function value, analogous to nffv 5548. To prove a deduction version of this analogous to nffvd 5550 is not easily possible because a deduction version of nfdfat 28098 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ x ( F''' A )
 
Theoremcsbafv12g 28105 Move class substitution in and out of a function value, analogous to csbfv12g 5551, with a direct proof proposed by Mario Carneiro, analogous to csbovg 5905. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ ( F''' B )  =  (
 [_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
 
Theoremafvfundmfveq 28106 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 28107 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 28108 The value of a class outside its domain is the universe, compare with ndmfv 5568. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  A  e.  dom  F  ->  ( F''' A )  =  _V )
 
Theoremafvvdm 28109 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 28110 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5574 (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  Fun  ( F  |`  { A } )  ->  ( F''' A )  =  _V )
 
Theoremafvvfunressn 28111 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 28112 A function's value at a proper class is the universe, compare with fvprc 5535. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  A  e.  _V  ->  ( F''' A )  =  _V )
 
Theoremafvvv 28113 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 28114 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 28115 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 28116 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 28117 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 28118 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 28119 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 28120 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 28121* The value of a function at a unique point, analogous to fveu 5533. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  ( E! x  A F x  ->  ( F''' A )  =  U. { x  |  A F x }
 )
 
Theoremfnbrafvb 28122 Equivalence of function value and binary relation, analogous to fnbrfvb 5579. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( ( F''' B )  =  C  <->  B F C ) )
 
Theoremfnopafvb 28123 Equivalence of function value and ordered pair membership, analogous to fnopfvb 5580. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( ( F''' B )  =  C  <->  <. B ,  C >.  e.  F ) )
 
Theoremfunbrafvb 28124 Equivalence of function value and binary relation, analogous to funbrfvb 5581. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( ( F''' A )  =  B  <->  A F B ) )
 
Theoremfunopafvb 28125 Equivalence of function value and ordered pair membership, analogous to funopfvb 5582. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( ( F''' A )  =  B  <->  <. A ,  B >.  e.  F ) )
 
Theoremfunbrafv 28126 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 5577. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fun  F  ->  ( A F B  ->  ( F''' A )  =  B ) )
 
Theoremfunbrafv2b 28127 Function value in terms of a binary relation, analogous to funbrfv2b 5583. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fun  F  ->  ( A F B  <->  ( A  e.  dom 
 F  /\  ( F''' A )  =  B ) ) )
 
Theoremdfafn5a 28128* Representation of a function in terms of its values, analogous to dffn5 5584 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
 
Theoremdfafn5b 28129* Representation of a function in terms of its values, analogous to dffn5 5584 (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 28130* The range of a function expressed as a collection of the function's values, analogous to fnrnfv 5585. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F''' x ) } )
 
Theoremafvelrnb 28131* A member of a function's range is a value of the function, analogous to fvelrnb 5586 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 28132* A member of a function's range is a value of the function, only one direction of implication of fvelrnb 5586. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
 |-  ( F  Fn  A  ->  ( B  e.  ran  F  ->  E. x  e.  A  ( F''' x )  =  B ) )
 
Theoremdfaimafn 28133* Alternate definition of the image of a function, analogous to dfimafn 5587. (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 28134* Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5588. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  C_  dom  F )  ->  ( F " A )  =  U_ x  e.  A  { ( F''' x ) } )
 
Theoremafvelima 28135* Function value in an image, analogous to fvelima 5590. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  ( F " B ) )  ->  E. x  e.  B  ( F''' x )  =  A )
 
Theoremafvelrn 28136 A function's value belongs to its range, analogous to fvelrn 5677. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( F''' A )  e.  ran  F )
 
Theoremfnafvelrn 28137 A function's value belongs to its range, analogous to fnfvelrn 5678. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( F''' B )  e.  ran  F )
 
Theoremfafvelrn 28138 A function's value belongs to its codomain, analogous to ffvelrn 5679. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F : A --> B  /\  C  e.  A )  ->  ( F''' C )  e.  B )
 
Theoremffnafv 28139* A function maps to a class to which all values belong, analogous to ffnfv 5701. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F''' x )  e.  B ) )
 
Theoremafvres 28140 The value of a restricted function, analogous to fvres 5558. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  e.  B  ->  ( ( F  |`  B )''' A )  =  ( F''' A ) )
 
Theoremtz6.12-afv 28141* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12 5561. (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 28142* Function value (Theorem 6.12(1) of [TakeutiZaring] p. 27, analogous to tz6.12-1 5560. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
 |-  (
 ( A F y 
 /\  E! y  A F y )  ->  ( F''' A )  =  y
 )
 
Theoremdmfcoafv 28143 Domains of a function composition, analogous to dmfco 5609. (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 28144 Value of a function composition, analogous to fvco2 5610. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  (
 ( G  Fn  A  /\  X  e.  A ) 
 ->  ( ( F  o.  G )''' X )  =  ( F''' ( G''' X ) ) )
 
Theoremrlimdmafv 28145 Two ways to express that a function has a limit, analogous to rlimdm 12041. (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 ) ) )
 
18.23.2.8  Alternative definition of the value of an operation
 
Theoremaoveq123d 28146 Equality deduction for operation value, analogous to oveq123d 5895. (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 28147 Bound-variable hypothesis builder for operation value, analogous to nfov 5897. To prove a deduction version of this analogous to nfovd 5896 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 28104). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x (( A F B))
 
Theoremcsbaovg 28148 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 28149 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 28150 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 28151 The value of an operation outside its domain, analogous to ndmafv 28108. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  <. A ,  B >.  e.  dom  F  -> (( A F B))  =  _V )
 
Theoremndmaovg 28152 The value of an operation outside its domain, analogous to ndmovg 6019. (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 28153 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 28154 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 28155 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 28156 The value of an operation when the one of the arguments is a proper class, analogous to ovprc 5901. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  Rel  dom 
 F   =>    |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> (( A F B))  =  _V )
 
Theoremaovrcl 28157 Reverse closure for an operation value, analogous to afvvv 28113. In contrast to ovrcl 5904, 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 28158 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 28159 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 28160 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 28161 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 28162 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 28163 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 28164 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 28165* A frequently used special case of rspc2ev 2905 for operation values, analogous to rspceov 5909. (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 28166 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5580. (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 28167* An operation maps to a class to which all values belong, analogous to ffnov 5964. (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 28168 Closure law for an operation, analogous to fovcl 5965. (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 28169* Value of a function given by the "maps to" notation, analogous to ovmpt4g 5986. (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 28170* Domain of closure of an operation. In contrast to oprssdm 6018, 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 28171 The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6021 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 28172 Reverse closure law, in contrast to ndmovrcl 6022 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 28173 Any operation is commutative outside its domain, analogous to ndmovcom 6023. (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 28174 Any operation is associative outside its domain. In contrast to ndmovass 6024 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 28175 Any operation is distributive outside its domain. In contrast to ndmovdistr 6025 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 (25-Nov-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 23878), for the degree of a vertex (df-vdgr 23880) and for Eulerian Paths (df-eupa 23879). These definitions (and the corresponding theorems) are not used within this section, except for the proofs of the compatibility between the definitions (see uslisumgra 28246, usisumgra 28248 and eupatrl 28385).

After providing some additional theorems of set theory (which will be used in the proofs of the theorems for graph theory), the definitions and theorems for graph theory start in subsubsection "Undirected simple graphs" (with syntax definition cuslg 28226).

 
18.23.3.1  Logical disjunction and conjunction (extension)
 
Theoremjaoi2 28176 If one part of a disjunction is already true, the other part can be true only if the first part is false. (Contributed by Alexander van der Vekens, 3-Nov-2017.)
 |-  (
 ( ph  \/  ( -.  ph  /\  ch )
 )  ->  ps )   =>    |-  (
 ( ph  \/  ch )  ->  ps )
 
18.23.3.2  Abbreviated conjunction and disjunction of three wff's (extension)
 
Theorem3bior1fd 28177 A wff is equivalent to its threefold disjunction with single falsehood, analogous to biorf 394. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
 |-  ( ph  ->  -.  th )   =>    |-  ( ph  ->  ( ( ch 
 \/  ps )  <->  ( th  \/  ch 
 \/  ps ) ) )
 
Theorem3bior1fand 28178 A wff is equivalent to its threefold disjunction with single falsehood of a conjunction. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
 |-  ( ph  ->  -.  th )   =>    |-  ( ph  ->  ( ( ch 
 \/  ps )  <->  ( ( th  /\ 
 ta )  \/  ch  \/  ps ) ) )
 
Theorem3bior2fd 28179 A wff is equivalent to its threefold disjunction with double falsehood, analogous to biorf 394. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
 |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  ( ps  <->  ( th  \/  ch 
 \/  ps ) ) )
 
Theorem3biant1d 28180 A wff is equivalent to its threefold conjunction with single truth, analogous to biantrud 493. (Contributed by Alexander van der Vekens, 26-Sep-2017.)
 |-  ( ph  ->  th )   =>    |-  ( ph  ->  (
 ( ch  /\  ps ) 
 <->  ( th  /\  ch  /\ 
 ps ) ) )
 
18.23.3.3  Unordered and ordered pairs (extension)
 
Theoremtppreq3 28181 An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B }
 )
 
Theoremtpprceq3 28182 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 28183 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 28184 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 ) )
 
Theoremdisjpr2 28185 The intersection of distinct pairs is disjoint, analogous to disjsn2 3707. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  (
 ( ( A  =/=  C 
 /\  B  =/=  C )  /\  ( A  =/=  D 
 /\  B  =/=  D ) )  ->  ( { A ,  B }  i^i  { C ,  D } )  =  (/) )
 
18.23.3.4  Functions (extension)
 
Theoremf1oprg 28186 An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 5531. (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 28187 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 } ) ) )
 
Theoremf1veqaeq 28188 If the values of the converse of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  (
 ( F : A -1-1-> B 
 /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( ( F `  C )  =  ( F `  D )  ->  C  =  D ) )
 
Theoremf1ocnvfvrneq 28189 If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  (
 ( F : A -1-1-> B 
 /\  ( C  e.  ran 
 F  /\  D  e.  ran 
 F ) )  ->  ( ( `' F `  C )  =  ( `' F `  D ) 
 ->  C  =  D ) )
 
Theoremopabex3d 28190* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ( ph  /\  x  e.  A )  ->  { y  |  ps }  e.  _V )   =>    |-  ( ph  ->  { <. x ,  y >.  |  ( x  e.  A  /\  ps ) }  e.  _V )
 
Theoremopabbrex 28191* A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  (
 ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V W E ) p 
 ->  th ) )   &    |-  (
 ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  th }  e.  _V )   =>    |-  ( ( V  e.  _V 
 /\  E  e.  _V )  ->  { <. f ,  p >.  |  (
 f ( V W E ) p  /\  ps ) }  e.  _V )
 
Theorem0neqopab 28192 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
 |-  -.  (/) 
 e.  { <. x ,  y >.  |  ph }
 
Theorembrabv 28193 If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
 |-  ( X { <. x ,  y >.  |  ph } Y  ->  ( X  e.  _V  /\  Y  e.  _V )
 )
 
18.23.3.5  Operations (Extension)
 
Theoremnssdmovg 28194 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 )  =  (/) )
 
18.23.3.6  "Maps to" notation (Extension)

The following theorems are about maps-to operations ( see df-mpt2 5879) 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 5940, ovmpt2x 5992 and fmpt2x 6206). 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 28195* 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 28196* 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 28197* 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 28198* 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 28199* 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 28200* 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 )  =  (/) )
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