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Theorem List for Intuitionistic Logic Explorer - 5601-5700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1ofveu 5601* There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  E! x  e.  A  ( F `  x )  =  C )
 
Theoremf1ocnvfv3 5602* Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( F : A
 -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  =  (
 iota_ x  e.  A  ( F `  x )  =  C ) )
 
Theoremriotaund 5603* Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)
 |-  ( -.  E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  =  (/) )
 
Theoremacexmidlema 5604* Lemma for acexmid 5612. (Contributed by Jim Kingdon, 6-Aug-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( { (/) }  e.  A  -> 
 ph )
 
Theoremacexmidlemb 5605* Lemma for acexmid 5612. (Contributed by Jim Kingdon, 6-Aug-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( (/) 
 e.  B  ->  ph )
 
Theoremacexmidlemph 5606* Lemma for acexmid 5612. (Contributed by Jim Kingdon, 6-Aug-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( ph  ->  A  =  B )
 
Theoremacexmidlemab 5607* Lemma for acexmid 5612. (Contributed by Jim Kingdon, 6-Aug-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  (
 ( ( iota_ v  e.  A  E. u  e.  y  ( A  e.  u  /\  v  e.  u ) )  =  (/)  /\  ( iota_
 v  e.  B  E. u  e.  y  ( B  e.  u  /\  v  e.  u )
 )  =  { (/) } )  ->  -.  ph )
 
Theoremacexmidlemcase 5608* Lemma for acexmid 5612. Here we divide the proof into cases (based on the disjunction implicit in an unordered pair, not the sort of case elimination which relies on excluded middle).

The cases are (1) the choice function evaluated at  A equals  { (/) }, (2) the choice function evaluated at  B equals  (/), and (3) the choice function evaluated at  A equals 
(/) and the choice function evaluated at  B equals  { (/) }.

Because of the way we represent the choice function  y, the choice function evaluated at  A is  ( iota_ v  e.  A E. u  e.  y ( A  e.  u  /\  v  e.  u ) ) and the choice function evaluated at  B is  ( iota_ v  e.  B E. u  e.  y ( B  e.  u  /\  v  e.  u ) ). Other than the difference in notation these work just as  ( y `  A ) and  ( y `  B ) would if  y were a function as defined by df-fun 4983.

Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at  A equals  { (/) }, then  { (/) }  e.  A and likewise for  B.

(Contributed by Jim Kingdon, 7-Aug-2019.)

 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( A. z  e.  C  E! v  e.  z  E. u  e.  y  ( z  e.  u  /\  v  e.  u )  ->  ( { (/) }  e.  A  \/  (/)  e.  B  \/  ( ( iota_ v  e.  A  E. u  e.  y  ( A  e.  u  /\  v  e.  u ) )  =  (/)  /\  ( iota_
 v  e.  B  E. u  e.  y  ( B  e.  u  /\  v  e.  u )
 )  =  { (/) } )
 ) )
 
Theoremacexmidlem1 5609* Lemma for acexmid 5612. List the cases identified in acexmidlemcase 5608 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( A. z  e.  C  E! v  e.  z  E. u  e.  y  ( z  e.  u  /\  v  e.  u )  ->  ( ph  \/  -.  ph ) )
 
Theoremacexmidlem2 5610* Lemma for acexmid 5612. This builds on acexmidlem1 5609 by noting that every element of  C is inhabited.

(Note that  y is not quite a function in the df-fun 4983 sense because it uses ordered pairs as described in opthreg 4345 rather than df-op 3440).

The set  A is also found in onsucelsucexmidlem 4318.

(Contributed by Jim Kingdon, 5-Aug-2019.)

 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }   &    |-  B  =  { x  e.  { (/) ,  { (/)
 } }  |  ( x  =  { (/) }  \/  ph ) }   &    |-  C  =  { A ,  B }   =>    |-  ( A. z  e.  C  A. w  e.  z  E! v  e.  z  E. u  e.  y  (
 z  e.  u  /\  v  e.  u )  ->  ( ph  \/  -.  ph ) )
 
Theoremacexmidlemv 5611* Lemma for acexmid 5612.

This is acexmid 5612 with additional distinct variable constraints, most notably between  ph and  x.

(Contributed by Jim Kingdon, 6-Aug-2019.)

 |- 
 E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  ( z  e.  u  /\  v  e.  u )   =>    |-  ( ph  \/  -.  ph )
 
Theoremacexmid 5612* The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer] p. 483.

The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function  y provides a value when  z is inhabited (as opposed to nonempty as in some statements of the axiom of choice).

Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967).

(Contributed by Jim Kingdon, 4-Aug-2019.)

 |- 
 E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  ( z  e.  u  /\  v  e.  u )   =>    |-  ( ph  \/  -.  ph )
 
2.6.10  Operations
 
Syntaxco 5613 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.
 class  ( A F B )
 
Syntaxcoprab 5614 Extend class notation to include class abstraction (class builder) of nested ordered pairs.
 class  { <. <. x ,  y >. ,  z >.  |  ph }
 
Syntaxcmpt2 5615 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 5616 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 . 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 5642 and ovprc2 5643. On the other hand, we often find uses for this definition when  F is a proper class.  F is normally equal to a class of nested ordered pairs of the form defined by df-oprab 5617. (Contributed by NM, 28-Feb-1995.)
 |-  ( A F B )  =  ( F ` 
 <. A ,  B >. )
 
Definitiondf-oprab 5617* 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 5616 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 5737. (Contributed by NM, 12-Mar-1995.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
 <. <. x ,  y >. ,  z >.  /\  ph ) }
 
Definitiondf-mpt2 5618* 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 3876 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 5619 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( F  =  G  ->  ( A F B )  =  ( A G B ) )
 
Theoremoveq1 5620 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( A  =  B  ->  ( A F C )  =  ( B F C ) )
 
Theoremoveq2 5621 Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  ( A  =  B  ->  ( C F A )  =  ( C F B ) )
 
Theoremoveq12 5622 Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A F C )  =  ( B F D ) )
 
Theoremoveq1i 5623 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  A  =  B   =>    |-  ( A F C )  =  ( B F C )
 
Theoremoveq2i 5624 Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
 |-  A  =  B   =>    |-  ( C F A )  =  ( C F B )
 
Theoremoveq12i 5625 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 5626 Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
 |-  A  =  B   =>    |-  ( C A D )  =  ( C B D )
 
Theoremoveq123i 5627 Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
 |-  A  =  C   &    |-  B  =  D   &    |-  F  =  G   =>    |-  ( A F B )  =  ( C G D )
 
Theoremoveq1d 5628 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A F C )  =  ( B F C ) )
 
Theoremoveq2d 5629 Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C F A )  =  ( C F B ) )
 
Theoremoveqd 5630 Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C A D )  =  ( C B D ) )
 
Theoremoveq12d 5631 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 5632 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 5633 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 5634 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 5635 Deduction version of bound-variable hypothesis builder nfov 5636. (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 5636 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 )
 
Theoremoprabidlem 5637* Slight elaboration of exdistrfor 1725. A lemma for oprabid 5638. (Contributed by Jim Kingdon, 15-Jan-2019.)
 |-  ( E. x E. y ( x  =  z  /\  ps )  ->  E. x ( x  =  z  /\  E. y ps ) )
 
Theoremoprabid 5638 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between  x,  y, and  z, we use ax-bndl 1442 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  ( <. <. x ,  y >. ,  z >.  e.  { <.
 <. x ,  y >. ,  z >.  |  ph }  <->  ph )
 
Theoremfnovex 5639 The result of an operation is a set. (Contributed by Jim Kingdon, 15-Jan-2019.)
 |-  ( ( F  Fn  ( C  X.  D ) 
 /\  A  e.  C  /\  B  e.  D ) 
 ->  ( A F B )  e.  _V )
 
Theoremovexg 5640 Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
 |-  ( ( A  e.  V  /\  F  e.  W  /\  B  e.  X ) 
 ->  ( A F B )  e.  _V )
 
Theoremovprc 5641 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 5642 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 5643 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 )  =  (/) )
 
Theoremcsbov123g 5644 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 5645* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C ) )
 
Theoremcsbov1g 5646* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F C ) )
 
Theoremcsbov2g 5647* Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B F C )  =  ( B F [_ A  /  x ]_ C ) )
 
Theoremrspceov 5648* A frequently used special case of rspc2ev 2728 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 5649 Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5309. (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 ) )
 
Theoremopabbrex 5650* 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 5651 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 }
 
Theorembrabvv 5652* If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Jim Kingdon, 16-Jan-2019.)
 |-  ( X { <. x ,  y >.  |  ph } Y  ->  ( X  e.  _V  /\  Y  e.  _V ) )
 
Theoremdfoprab2 5653* 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 5654* An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
 |- 
 Rel  { <. <. x ,  y >. ,  z >.  |  ph }
 
Theoremnfoprab1 5655 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 5656 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 5657 The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
 |-  F/_ z { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremnfoprab 5658* 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 5659* 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 } )
 
Theoremoprabbidv 5660* Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  {
 <. <. x ,  y >. ,  z >.  |  ps }  =  { <. <. x ,  y >. ,  z >.  |  ch } )
 
Theoremoprabbii 5661* Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( ph  <->  ps )   =>    |- 
 { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z >.  |  ps }
 
Theoremssoprab2 5662 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4076. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( A. x A. y A. z ( ph  ->  ps )  ->  { <. <. x ,  y >. ,  z >.  |  ph }  C_  {
 <. <. x ,  y >. ,  z >.  |  ps } )
 
Theoremssoprab2b 5663 Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4077. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( { <. <. x ,  y >. ,  z >.  | 
 ph }  C_  { <. <. x ,  y >. ,  z >.  |  ps }  <->  A. x A. y A. z ( ph  ->  ps ) )
 
Theoremeqoprab2b 5664 Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4080. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  y >. ,  z >.  |  ps }  <->  A. x A. y A. z ( ph  <->  ps ) )
 
Theoremmpt2eq123 5665* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)
 |-  ( ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F ) )
 
Theoremmpt2eq12 5666* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  ( x  e.  A ,  y  e.  B  |->  E )  =  ( x  e.  C ,  y  e.  D  |->  E ) )
 
Theoremmpt2eq123dva 5667* An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  =  E )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  C  =  F )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F ) )
 
Theoremmpt2eq123dv 5668* An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ph  ->  B  =  E )   &    |-  ( ph  ->  C  =  F )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F ) )
 
Theoremmpt2eq123i 5669 An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.)
 |-  A  =  D   &    |-  B  =  E   &    |-  C  =  F   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )
 
Theoremmpt2eq3dva 5670* Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.)
 |-  ( ( ph  /\  x  e.  A  /\  y  e.  B )  ->  C  =  D )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D ) )
 
Theoremmpt2eq3ia 5671 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
 
Theoremnfmpt21 5672 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
 |-  F/_ x ( x  e.  A ,  y  e.  B  |->  C )
 
Theoremnfmpt22 5673 Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
 |-  F/_ y ( x  e.  A ,  y  e.  B  |->  C )
 
Theoremnfmpt2 5674* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
 |-  F/_ z A   &    |-  F/_ z B   &    |-  F/_ z C   =>    |-  F/_ z ( x  e.  A ,  y  e.  B  |->  C )
 
Theoremmpt20 5675 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)
 
Theoremoprab4 5676* Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }
 
Theoremcbvoprab1 5677* Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
 |- 
 F/ w ph   &    |-  F/ x ps   &    |-  ( x  =  w  ->  (
 ph 
 <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. w ,  y >. ,  z >.  |  ps }
 
Theoremcbvoprab2 5678* Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |- 
 F/ w ph   &    |-  F/ y ps   &    |-  ( y  =  w  ->  ( ph  <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  w >. ,  z >.  |  ps }
 
Theoremcbvoprab12 5679* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |- 
 F/ w ph   &    |-  F/ v ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph 
 <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. w ,  v >. ,  z >.  |  ps }
 
Theoremcbvoprab12v 5680* Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)
 |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph 
 <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. w ,  v >. ,  z >.  |  ps }
 
Theoremcbvoprab3 5681* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
 |- 
 F/ w ph   &    |-  F/ z ps   &    |-  ( z  =  w  ->  ( ph  <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  y >. ,  w >.  |  ps }
 
Theoremcbvoprab3v 5682* Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( z  =  w 
 ->  ( ph  <->  ps ) )   =>    |-  { <. <. x ,  y >. ,  z >.  | 
 ph }  =  { <.
 <. x ,  y >. ,  w >.  |  ps }
 
Theoremcbvmpt2x 5683* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5684 allows  B to be a function of  x. (Contributed by NM, 29-Dec-2014.)
 |-  F/_ z B   &    |-  F/_ x D   &    |-  F/_ z C   &    |-  F/_ w C   &    |-  F/_ x E   &    |-  F/_ y E   &    |-  ( x  =  z 
 ->  B  =  D )   &    |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  E )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  A ,  w  e.  D  |->  E )
 
Theoremcbvmpt2 5684* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.)
 |-  F/_ z C   &    |-  F/_ w C   &    |-  F/_ x D   &    |-  F/_ y D   &    |-  ( ( x  =  z  /\  y  =  w )  ->  C  =  D )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  A ,  w  e.  B  |->  D )
 
Theoremcbvmpt2v 5685* Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3908, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)
 |-  ( x  =  z 
 ->  C  =  E )   &    |-  ( y  =  w  ->  E  =  D )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  (
 z  e.  A ,  w  e.  B  |->  D )
 
Theoremdmoprab 5686* The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |- 
 dom  { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. x ,  y >.  |  E. z ph }
 
Theoremdmoprabss 5687* The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theoremrnoprab 5688* The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)
 |- 
 ran  { <. <. x ,  y >. ,  z >.  |  ph }  =  { z  | 
 E. x E. y ph }
 
Theoremrnoprab2 5689* The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)
 |- 
 ran  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  =  { z  |  E. x  e.  A  E. y  e.  B  ph }
 
Theoremreldmoprab 5690* The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.)
 |- 
 Rel  dom  { <. <. x ,  y >. ,  z >.  | 
 ph }
 
Theoremoprabss 5691* Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
 |- 
 { <. <. x ,  y >. ,  z >.  |  ph } 
 C_  ( ( _V 
 X.  _V )  X.  _V )
 
Theoremeloprabga 5692* The law of concretion for operation class abstraction. Compare elopab 4059. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y >. ,  z >.  |  ph }  <->  ps ) )
 
Theoremeloprabg 5693* The law of concretion for operation class abstraction. Compare elopab 4059. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 y  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  C  ->  ( ch  <->  th ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y >. ,  z >.  |  ph }  <->  th ) )
 
Theoremssoprab2i 5694* Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  ( ph  ->  ps )   =>    |-  { <. <. x ,  y >. ,  z >.  |  ph }  C_  {
 <. <. x ,  y >. ,  z >.  |  ps }
 
Theoremmpt2v 5695* Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
 |-  ( x  e.  _V ,  y  e.  _V  |->  C )  =  { <.
 <. x ,  y >. ,  z >.  |  z  =  C }
 
Theoremmpt2mptx 5696* Express a two-argument function as a one-argument function, or vice-versa. In this version 
B ( x ) is not assumed to be constant w.r.t  x. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( z  =  <. x ,  y >.  ->  C  =  D )   =>    |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
 
Theoremmpt2mpt 5697* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
 |-  ( z  =  <. x ,  y >.  ->  C  =  D )   =>    |-  ( z  e.  ( A  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
 
Theoremresoprab 5698* Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
 |-  ( { <. <. x ,  y >. ,  z >.  | 
 ph }  |`  ( A  X.  B ) )  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }
 
Theoremresoprab2 5699* Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( C  C_  A  /\  D  C_  B )  ->  ( { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  A  /\  y  e.  B )  /\  ph ) }  |`  ( C  X.  D ) )  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  C  /\  y  e.  D )  /\  ph ) } )
 
Theoremresmpt2 5700* Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
 |-  ( ( C  C_  A  /\  D  C_  B )  ->  ( ( x  e.  A ,  y  e.  B  |->  E )  |`  ( C  X.  D ) )  =  ( x  e.  C ,  y  e.  D  |->  E ) )
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