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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfnresfnco 27401 Composition of two functions, similar to fnco 5352. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( F  |`  ran  G )  Fn  ran  G  /\  G  Fn  B )  ->  ( F  o.  G )  Fn  B )
 
Theoremfuncoressn 27402 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( ( G `
  X )  e. 
 dom  F  /\  Fun  ( F  |`  { ( G `
  X ) }
 ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( ( F  o.  G )  |`  { X } ) )
 
Theoremfunressnfv 27403 A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( X  e.  dom  ( F  o.  G )  /\  Fun  ( ( F  o.  G )  |`  { X } ) ) 
 /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( F  |`  { ( G `
  X ) }
 ) )
 
18.23.2.5  Predicate "defined at"
 
Theoremdfateq12d 27404 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F defAt  A  <->  G defAt  B ) )
 
Theoremnfdfat 27405 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g. for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |- 
 F/ x  F defAt  A
 
Theoremdfdfat2 27406* Alternate definition of the predicate "defined at" not using the  Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F defAt  A  <->  ( A  e.  dom 
 F  /\  E! y  A F y ) )
 
18.23.2.6  Alternative definition of the value of a function
 
Theoremdfafv2 27407 Alternative definition of  ( F''' A ) using  ( F `
 A ) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F''' A )  =  if ( F defAt  A ,  ( F `  A ) ,  _V )
 
Theoremafveq12d 27408 Equality deduction for function value, analogous to fveq12d 5531. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F''' A )  =  ( G''' B ) )
 
Theoremafveq1 27409 Equality theorem for function value, analogous to fveq1 5524. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F  =  G  ->  ( F''' A )  =  ( G''' A ) )
 
Theoremafveq2 27410 Equality theorem for function value, analogous to fveq1 5524. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  =  B  ->  ( F''' A )  =  ( F''' B ) )
 
Theoremnfafv 27411 Bound-variable hypothesis builder for function value, analogous to nffv 5532. To prove a deduction version of this analogous to nffvd 5534 is not easily possible because a deduction version of nfdfat 27405 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ x ( F''' A )
 
Theoremcsbafv12g 27412 Move class substitution in and out of a function value, analogous to csbfv12g 5535, with a direct proof proposed by Mario Carneiro, analogous to csbovg 5889. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ ( F''' B )  =  (
 [_ A  /  x ]_ F''' [_ A  /  x ]_ B ) )
 
Theoremafvfundmfveq 27413 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 27414 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 27415 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 27416 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 27417 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 27418 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 27419 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 27420 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 27421 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 27422 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 27423 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 27424 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 27425 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 27426 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 27427 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 27428 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 27429 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 27430 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 27431 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 27432 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 27433 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 27434* 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 27435* 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 27436* 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 27437* 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 27438* 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 27439* 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 27440* 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 27441* 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 27442 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 27443 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 27444 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 27445* 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 27446 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 27447* 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 27448 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 27449 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 27450 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 27451 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 27411). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x (( A F B))
 
Theoremcsbaovg 27452 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 27453 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 27454 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 27455 The value of an operation outside its domain, analogous to ndmafv 27415. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  <. A ,  B >.  e.  dom  F  -> (( A F B))  =  _V )
 
Theoremndmaovg 27456 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 27457 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 27458 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 27459 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 27460 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 27461 Reverse closure for an operation value, analogous to afvvv 27420. 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 27462 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 27463 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 27464 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 27465 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 27466 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 27467 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 27468 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 27469* 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 27470 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 27471* 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 27472 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 27473* 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 27474* 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 27475 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 27476 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 27477 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 27478 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 27479 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 23274), for the degree of a vertex (df-vdgr 23276) and for Eulerian Paths (df-eupa 23275). These definitions (and the corresponding theorems) are not used within this section, except for the proofs of the compatibility between the definitions (see uslisumgra 27511 and usisumgra 27513).

 
18.23.3.1  Unordered and ordered pairs (extension)
 
Theoremdifprsneq 27480 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A } )
 
Theoremdiftpsneq 27481 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 27482 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 27483 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 27484 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 27485 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 27486 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 27487 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  Longer string literals (extension)
 
Theorems2prop 27488 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 27489 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 27490 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 27491 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, 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 27492 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 } ) ) ) )
 
18.23.3.4  Undirected simple graphs

Although undirected simple graphs (with or without loops) are defined separately from undirected mulitigraphs (see df-umgra 23274), the definitions are similar and therefore compatible with each other, see uslisumgra 27511 and usisuslgra 27512.

 
Syntaxcuslg 27493 Extend class notation with undirected (simple) graphs with loops.
 class USLGrph
 
Syntaxcusg 27494 Extend class notation with undirected (simple) graphs (without loops).
 class USGrph
 
Definitiondf-uslgra 27495* 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 27496* 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 27497 The class of all undirected simple graph with loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  Rel USLGrph
 
Theoremrelusgra 27498 The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  Rel USGrph
 
Theoremuslgrav 27499 The classes of vertices and edges of a class ar sets. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USLGrph  E  ->  ( V  e.  _V  /\  E  e.  _V ) )
 
Theoremusgrav 27500 The classes of vertices and edges of a class ar sets. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( V USGrph  E  ->  ( V  e.  _V  /\  E  e.  _V ) )
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