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Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcocan2 5701 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B )  ->  ( ( H  o.  F )  =  ( K  o.  F ) 
 <->  H  =  K ) )
 
Theoremfcof1o 5702 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ( ( F : A --> B  /\  G : B --> A ) 
 /\  ( ( F  o.  G )  =  (  _I  |`  B ) 
 /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F : A -1-1-onto-> B  /\  `' F  =  G ) )
 
Theoremfoeqcnvco 5703 Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  ( ( F : A -onto-> B  /\  G : A -onto-> B )  ->  ( F  =  G  <->  ( F  o.  `' G )  =  (  _I  |`  B )
 ) )
 
Theoremf1eqcocnv 5704 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  ( ( F : A -1-1-> B  /\  G : A -1-1-> B )  ->  ( F  =  G  <->  ( `' F  o.  G )  =  (  _I  |`  A )
 ) )
 
Theoremfveqf1o 5705 Given a bijection  F, produce another bijection  G which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  G  =  ( F  o.  ( (  _I  |`  ( A  \  { C ,  ( `' F `  D ) }
 ) )  u.  { <. C ,  ( `' F `  D )
 >. ,  <. ( `' F `  D ) ,  C >. } ) )   =>    |-  ( ( F : A -1-1-onto-> B  /\  C  e.  A  /\  D  e.  B )  ->  ( G : A
 -1-1-onto-> B  /\  ( G `  C )  =  D ) )
 
Theoremfliftrel 5706*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  F  C_  ( R  X.  S ) )
 
Theoremfliftel 5707* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B ) ) )
 
Theoremfliftel1 5708* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ( ph  /\  x  e.  X )  ->  A F B )
 
Theoremfliftcnv 5709* Converse of the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  `' F  =  ran  (  x  e.  X  |->  <. B ,  A >. ) )
 
Theoremfliftfun 5710* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   &    |-  ( x  =  y  ->  A  =  C )   &    |-  ( x  =  y  ->  B  =  D )   =>    |-  ( ph  ->  ( Fun  F  <->  A. x  e.  X  A. y  e.  X  ( A  =  C  ->  B  =  D ) ) )
 
Theoremfliftfund 5711* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   &    |-  ( x  =  y  ->  A  =  C )   &    |-  ( x  =  y  ->  B  =  D )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  A  =  C ) )  ->  B  =  D )   =>    |-  ( ph  ->  Fun  F )
 
Theoremfliftfuns 5712* The function  F is the unique function defined by  F `  A  =  B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  ( Fun  F  <->  A. y  e.  X  A. z  e.  X  (
 [_ y  /  x ]_ A  =  [_ z  /  x ]_ A  ->  [_ y  /  x ]_ B  =  [_ z  /  x ]_ B ) ) )
 
Theoremfliftf 5713* The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   =>    |-  ( ph  ->  ( Fun  F  <->  F : ran  (  x  e.  X  |->  A ) --> S ) )
 
Theoremfliftval 5714* The value of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  (  x  e.  X  |->  <. A ,  B >. )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  R )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  S )   &    |-  ( x  =  Y  ->  A  =  C )   &    |-  ( x  =  Y  ->  B  =  D )   &    |-  ( ph  ->  Fun 
 F )   =>    |-  ( ( ph  /\  Y  e.  X )  ->  ( F `  C )  =  D )
 
Theoremisoeq1 5715 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  G  Isom  R ,  S  ( A ,  B ) ) )
 
Theoremisoeq2 5716 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( R  =  T  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  T ,  S  ( A ,  B ) ) )
 
Theoremisoeq3 5717 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( S  =  T  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  T  ( A ,  B ) ) )
 
Theoremisoeq4 5718 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( A  =  C  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  S  ( C ,  B ) ) )
 
Theoremisoeq5 5719 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
 |-  ( B  =  C  ->  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  S  ( A ,  C ) ) )
 
Theoremnfiso 5720 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  F/_ x H   &    |-  F/_ x R   &    |-  F/_ x S   &    |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  H  Isom  R ,  S  ( A ,  B )
 
Theoremisof1o 5721 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B )
 
Theoremisorel 5722 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C R D  <->  ( H `  C ) S ( H `  D ) ) )
 
Theoremsoisores 5723* Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ( ( R  Or  B  /\  S  Or  C )  /\  ( F : B --> C  /\  A  C_  B ) ) 
 ->  ( ( F  |`  A ) 
 Isom  R ,  S  ( A ,  ( F
 " A ) )  <->  A. x  e.  A  A. y  e.  A  ( x R y  ->  ( F `  x ) S ( F `  y ) ) ) )
 
Theoremsoisoi 5724* Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( ( ( R  Or  A  /\  S  Po  B )  /\  ( H : A -onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  ->  ( H `  x ) S ( H `  y
 ) ) ) ) 
 ->  H  Isom  R ,  S  ( A ,  B ) )
 
Theoremisoid 5725 Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
 |-  (  _I  |`  A ) 
 Isom  R ,  R  ( A ,  A )
 
Theoremisocnv 5726 Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
 
Theoremisocnv2 5727 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  `' R ,  `' S ( A ,  B ) )
 
Theoremisocnv3 5728 Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  C  =  ( ( A  X.  A ) 
 \  R )   &    |-  D  =  ( ( B  X.  B )  \  S )   =>    |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  C ,  D  ( A ,  B ) )
 
Theoremisores2 5729 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  R ,  ( S  i^i  ( B  X.  B ) ) ( A ,  B ) )
 
Theoremisores1 5730 An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( H  Isom  R ,  S  ( A ,  B ) 
 <->  H  Isom  ( R  i^i  ( A  X.  A ) ) ,  S ( A ,  B ) )
 
Theoremisores3 5731 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  K  C_  A  /\  X  =  ( H " K ) )  ->  ( H  |`  K )  Isom  R ,  S  ( K ,  X ) )
 
Theoremisotr 5732 Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  G  Isom  S ,  T  ( B ,  C ) )  ->  ( G  o.  H )  Isom  R ,  T  ( A ,  C ) )
 
Theoremisomin 5733 Isomorphisms preserve minimal elements. Note that  ( `' R " { D } ) is Takeuti and Zaring's idiom for the initial segment  { x  |  x R D }. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A )
 )  ->  ( ( C  i^i  ( `' R " { D } )
 )  =  (/)  <->  ( ( H
 " C )  i^i  ( `' S " { ( H `  D ) } )
 )  =  (/) ) )
 
Theoremisoini 5734 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  ( H " ( A  i^i  ( `' R " { D } )
 ) )  =  ( B  i^i  ( `' S " { ( H `  D ) }
 ) ) )
 
Theoremisoini2 5735 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
 |-  C  =  ( A  i^i  ( `' R " { X } )
 )   &    |-  D  =  ( B  i^i  ( `' S " { ( H `  X ) } )
 )   =>    |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  X  e.  A )  ->  ( H  |`  C ) 
 Isom  R ,  S  ( C ,  D ) )
 
Theoremisofrlem 5736* Lemma for isofr 5738. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  ( H " x )  e.  _V )   =>    |-  ( ph  ->  ( S  Fr  B  ->  R  Fr  A ) )
 
Theoremisoselem 5737* Lemma for isose 5739. (Contributed by Mario Carneiro, 23-Jun-2015.)
 |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  ( H " x )  e.  _V )   =>    |-  ( ph  ->  ( R Se  A  ->  S Se  B ) )
 
Theoremisofr 5738 An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A 
 <->  S  Fr  B ) )
 
Theoremisose 5739 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  <->  S Se 
 B ) )
 
Theoremisofr2 5740 A weak form of isofr 5738 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  B  e.  V )  ->  ( S  Fr  B  ->  R  Fr  A ) )
 
Theoremisopolem 5741 Lemma for isopo 5742. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( S  Po  B  ->  R  Po  A ) )
 
Theoremisopo 5742 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( R  Po  A 
 <->  S  Po  B ) )
 
Theoremisosolem 5743 Lemma for isoso 5744. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A ) )
 
Theoremisoso 5744 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( R  Or  A 
 <->  S  Or  B ) )
 
Theoremisowe 5745 An isomorphism preserves well ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  ( H  Isom  R ,  S  ( A ,  B )  ->  ( R  We  A 
 <->  S  We  B ) )
 
Theoremisowe2 5746* A weak form of isowe 5745 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  A. x ( H " x )  e.  _V )  ->  ( S  We  B  ->  R  We  A ) )
 
Theoremf1oiso 5747* Any one-to-one onto function determines an isomorphism with an induced relation  S. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.)
 |-  ( ( H : A
 -1-1-onto-> B  /\  S  =  { <. z ,  w >.  | 
 E. x  e.  A  E. y  e.  A  ( ( z  =  ( H `  x )  /\  w  =  ( H `  y ) )  /\  x R y ) } )  ->  H  Isom  R ,  S  ( A ,  B ) )
 
Theoremf1oiso2 5748* Any one-to-one onto function determines an isomorphism with an induced relation  S. (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  ( `' H `  x ) R ( `' H `  y ) ) }   =>    |-  ( H : A -1-1-onto-> B  ->  H  Isom  R ,  S  ( A ,  B ) )
 
Theoremf1owe 5749* Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)
 |-  R  =  { <. x ,  y >.  |  ( F `  x ) S ( F `  y ) }   =>    |-  ( F : A
 -1-1-onto-> B  ->  ( S  We  B  ->  R  We  A ) )
 
Theoremf1oweALT 5750* Well-ordering of isomorphic relations. (This version is proved directly instead of with the isomorphism predicate.) (Contributed by NM, 4-Mar-1997.) (Proof modification is discouraged.)
 |-  R  =  { <. x ,  y >.  |  ( F `  x ) S ( F `  y ) }   =>    |-  ( F : A
 -1-1-onto-> B  ->  ( S  We  B  ->  R  We  A ) )
 
Theoremweniso 5751 A set-like well-ordering has no nontrivial automorphisms. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( R  We  A  /\  R Se  A  /\  F  Isom  R ,  R  ( A ,  A ) )  ->  F  =  (  _I  |`  A )
 )
 
Theoremweisoeq 5752 Thus there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 5754. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( F 
 Isom  R ,  S  ( A ,  B ) 
 /\  G  Isom  R ,  S  ( A ,  B ) ) )  ->  F  =  G )
 
Theoremweisoeq2 5753 Thus there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 5755. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( ( S  We  B  /\  S Se  B )  /\  ( F 
 Isom  R ,  S  ( A ,  B ) 
 /\  G  Isom  R ,  S  ( A ,  B ) ) )  ->  F  =  G )
 
Theoremwemoiso 5754* Thus there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 7673. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( R  We  A  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
 
Theoremwemoiso2 5755* Thus there is at most one isomorphism between any two well-ordered sets. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( S  We  B  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
 
Theoremknatar 5756* The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice  ~P A. (Contributed by Mario Carneiro, 11-Jun-2015.)
 |-  X  =  |^| { z  e.  ~P A  |  ( F `  z ) 
 C_  z }   =>    |-  ( ( A  e.  V  /\  ( F `  A )  C_  A  /\  A. x  e. 
 ~P  A A. y  e.  ~P  x ( F `
  y )  C_  ( F `  x ) )  ->  ( X  C_  A  /\  ( F `
  X )  =  X ) )
 
2.4.8  Operations
 
Syntaxco 5757 Extend class notation to include the value of an operation  F (such as  +) for two arguments  A and  B. Note that the syntax is simply three class symbols in a row surrounded by parentheses. Since operation values are the only possible class expressions consisting of three class expressions in a row surrounded by parentheses, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 8971.)
 class  ( A F B )
 
Syntaxcoprab 5758 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
 class  { <. <. x ,  y >. ,  z >.  |  ph }
 
Syntaxcmpt2 5759 Extend the definition of a class to include maps-to notation for defining an operation via a rule.
 class  ( x  e.  A ,  y  e.  B  |->  C )
 
Definitiondf-ov 5760 Define the value of an operation. Definition of operation value in [Enderton] p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation  F and its arguments  A and  B- will be useful for proving meaningful theorems. For example, if class  F is the operation  + and arguments  A and  B are  3 and  2, the expression  ( 3  +  2 ) can be proved to equal  5 (see 3p2e5 9787). This definition is well-defined, although not very meaningful, when classes  A and/or  B are proper classes (i.e. are not sets); see ovprc1 5785 and ovprc2 5786. On the other hand, we often find uses for this definition when  F is a proper class, such as  +o in oav 6443.  F is normally equal to a class of nested ordered pairs of the form defined by df-oprab 5761. (Contributed by NM, 28-Feb-1995.)
 |-  ( A F B )  =  ( F ` 
 <. A ,  B >. )
 
Definitiondf-oprab 5761* Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally  x,  y, and  z are distinct, although the definition doesn't strictly require it. See df-ov 5760 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ovmpt2 5882. (Contributed by NM, 12-Mar-1995.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
 <. <. x ,  y >. ,  z >.  /\  ph ) }
 
Definitiondf-mpt2 5762* Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from  x ,  y (in  A  X.  B) to  B ( x ,  y )." An extension of df-mpt 4019 for two arguments. (Contributed by NM, 17-Feb-2008.)
 |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
 
Theoremoveq 5763 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( F  =  G  ->  ( A F B )  =  ( A G B ) )
 
Theoremoveq1 5764 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( A  =  B  ->  ( A F C )  =  ( B F C ) )
 
Theoremoveq2 5765 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( A  =  B  ->  ( C F A )  =  ( C F B ) )
 
Theoremoveq12 5766 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A F C )  =  ( B F D ) )
 
Theoremoveq1i 5767 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  A  =  B   =>    |-  ( A F C )  =  ( B F C )
 
Theoremoveq2i 5768 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  A  =  B   =>    |-  ( C F A )  =  ( C F B )
 
Theoremoveq12i 5769 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A F C )  =  ( B F D )
 
Theoremoveqi 5770 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
 |-  A  =  B   =>    |-  ( C A D )  =  ( C B D )
 
Theoremoveq123i 5771 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
 |-  A  =  C   &    |-  B  =  D   &    |-  F  =  G   =>    |-  ( A F B )  =  ( C G D )
 
Theoremoveq1d 5772 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A F C )  =  ( B F C ) )
 
Theoremoveq2d 5773 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C F A )  =  ( C F B ) )
 
Theoremoveqd 5774 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D )  =  ( C B D ) )
 
Theoremoveq12d 5775 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A F C )  =  ( B F D ) )
 
Theoremoveqan12d 5776 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A F C )  =  ( B F D ) )
 
Theoremoveqan12rd 5777 Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ps 
 /\  ph )  ->  ( A F C )  =  ( B F D ) )
 
Theoremoveq123d 5778 Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A F C )  =  ( B G D ) )
 
Theoremnfovd 5779 Deduction version of bound-variable hypothesis builder nfov 5780. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x F )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x ( A F B ) )
 
Theoremnfov 5780 Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  F/_ x B   =>    |-  F/_ x ( A F B )
 
Theoremoprabid 5781 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( <. <. x ,  y >. ,  z >.  e.  { <.
 <. x ,  y >. ,  z >.  |  ph }  <->  ph )
 
Theoremovex 5782 The result of an operation is a set. (Contributed by NM, 13-Mar-1995.)
 |-  ( A F B )  e.  _V
 
Theoremovssunirn 5783 The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( X F Y )  C_  U. ran  F
 
Theoremovprc 5784 The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 Rel  dom  F   =>    |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
 
Theoremovprc1 5785 The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.)
 |- 
 Rel  dom  F   =>    |-  ( -.  A  e.  _V 
 ->  ( A F B )  =  (/) )
 
Theoremovprc2 5786 The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |- 
 Rel  dom  F   =>    |-  ( -.  B  e.  _V 
 ->  ( A F B )  =  (/) )
 
Theoremovrcl 5787 Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)
 |- 
 Rel  dom  F   =>    |-  ( C  e.  ( A F B )  ->  ( A  e.  _V  /\  B  e.  _V )
 )
 
Theoremcsbovg 5788 Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
 |-  ( A  e.  D  -> 
 [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) )
 
Theoremcsbov12g 5789* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  D  -> 
 [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C ) )
 
Theoremcsbov1g 5790* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  D  -> 
 [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F C ) )
 
Theoremcsbov2g 5791* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  D  -> 
 [_ A  /  x ]_ ( B F C )  =  ( B F [_ A  /  x ]_ C ) )
 
Theoremrcla4eov 5792* A frequently used special case of rcla42ev 2843 for operation values. (Contributed by NM, 21-Mar-2007.)
 |-  ( ( C  e.  A  /\  D  e.  B  /\  S  =  ( C F D ) ) 
 ->  E. x  e.  A  E. y  e.  B  S  =  ( x F y ) )
 
Theoremfnotovb 5793 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5463. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  A  /\  D  e.  B ) 
 ->  ( ( C F D )  =  R  <->  <. C ,  D ,  R >.  e.  F ) )
 
Theoremdfoprab2 5794* Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. w ,  z >.  |  E. x E. y ( w  = 
 <. x ,  y >.  /\  ph ) }
 
Theoremreloprab 5795* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
 |- 
 Rel  { <. <. x ,  y >. ,  z >.  |  ph }
 
Theoremnfoprab1 5796 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  F/_ x { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremnfoprab2 5797 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 25-Apr-1995.) (Revised by David Abernethy, 30-Jul-2012.)
 |-  F/_ y { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremnfoprab3 5798 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
 |-  F/_ z { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremnfoprab 5799* Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
 |- 
 F/ w ph   =>    |-  F/_ w { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremoprabbid 5800* Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |- 
 F/ x ph   &    |-  F/ y ph   &    |-  F/ z ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  {
 <. <. x ,  y >. ,  z >.  |  ps }  =  { <. <. x ,  y >. ,  z >.  |  ch } )
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