P. 56 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5501-5600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	riotacl2 5501	Membership law for "the unique element in A such that ph." (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )
Theorem	riotacl 5502*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
Theorem	riotasbc 5503	Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph )
Theorem	riotabidva 5504*	Equivalent wff's yield equal restricted class abstractions (deduction rule). (rabbidva 2592 analog.) (Contributed by NM, 17-Jan-2012.)
|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )
Theorem	riotabiia 5505	Equivalent wff's yield equal restricted iotas (inference rule). (rabbiia 2591 analog.) (Contributed by NM, 16-Jan-2012.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ x e. A ps )
Theorem	riota1 5506*	Property of restricted iota. Compare iota1 4901. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )
Theorem	riota1a 5507	Property of iota. (Contributed by NM, 23-Aug-2011.)
|- ( ( x e. A /\ E! x e. A ph ) -> ( ph <-> ( iota x ( x e. A /\ ph ) ) = x ) )
Theorem	riota2df 5508*	A deduction version of riota2f 5509. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/ x ph   &    |- ( ph -> F/_ x B )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> B e. A )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   =>    |- ( ( ph /\ E! x e. A ps ) -> ( ch <-> ( iota_ x e. A ps ) = B ) )
Theorem	riota2f 5509*	This theorem shows a condition that allows us to represent a descriptor with a class expression B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x B   &    |- F/ x ps   &    |- ( x = B -> ( ph <-> ps ) )   =>    |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
Theorem	riota2 5510*	This theorem shows a condition that allows us to represent a descriptor with a class expression B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
|- ( x = B -> ( ph <-> ps ) )   =>    |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
Theorem	riotaprop 5511*	Properties of a restricted definite description operator. Todo (df-riota 5488 update): can some uses of riota2f 5509 be shortened with this? (Contributed by NM, 23-Nov-2013.)
|- F/ x ps   &    |- B = ( iota_ x e. A ph )   &    |- ( x = B -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph -> ( B e. A /\ ps ) )
Theorem	riota5f 5512*	A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> F/_ x B )   &    |- ( ph -> B e. A )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) )   =>    |- ( ph -> ( iota_ x e. A ps ) = B )
Theorem	riota5 5513*	A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.)
|- ( ph -> B e. A )   &    |- ( ( ph /\ x e. A ) -> ( ps <-> x = B ) )   =>    |- ( ph -> ( iota_ x e. A ps ) = B )
Theorem	riotass2 5514*	Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
|- ( ( ( A C_ B /\ A. x e. A ( ph -> ps ) ) /\ ( E. x e. A ph /\ E! x e. B ps ) ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ps ) )
Theorem	riotass 5515*	Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
|- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ph ) )
Theorem	moriotass 5516*	Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
|- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ph ) )
Theorem	snriota 5517	A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.)
|- ( E! x e. A ph -> { x e. A | ph } = { ( iota_ x e. A ph ) } )
Theorem	eusvobj2 5518*	Specify the same property in two ways when class B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- B e. _V   =>    |- ( E! x E. y e. A x = B -> ( E. y e. A x = B <-> A. y e. A x = B ) )
Theorem	eusvobj1 5519*	Specify the same object in two ways when class B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- B e. _V   =>    |- ( E! x E. y e. A x = B -> ( iota x E. y e. A x = B ) = ( iota x A. y e. A x = B ) )
Theorem	f1ofveu 5520*	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 )
Theorem	f1ocnvfv3 5521*	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 ) )
Theorem	riotaund 5522*	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 ) = (/) )
Theorem	acexmidlema 5523*	Lemma for acexmid 5531. (Contributed by Jim Kingdon, 6-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( { (/) } e. A -> ph )
Theorem	acexmidlemb 5524*	Lemma for acexmid 5531. (Contributed by Jim Kingdon, 6-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( (/) e. B -> ph )
Theorem	acexmidlemph 5525*	Lemma for acexmid 5531. (Contributed by Jim Kingdon, 6-Aug-2019.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ph ) }   &    |- C = { A , B }   =>    |- ( ph -> A = B )
Theorem	acexmidlemab 5526*	Lemma for acexmid 5531. (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 )
Theorem	acexmidlemcase 5527*	Lemma for acexmid 5531. 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 4924. 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 ) ) = { (/) } ) ) )
Theorem	acexmidlem1 5528*	Lemma for acexmid 5531. List the cases identified in acexmidlemcase 5527 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 ) )
Theorem	acexmidlem2 5529*	Lemma for acexmid 5531. This builds on acexmidlem1 5528 by noting that every element of C is inhabited. (Note that y is not quite a function in the df-fun 4924 sense because it uses ordered pairs as described in opthreg 4299 rather than df-op 3407). The set A is also found in onsucelsucexmidlem 4272. (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 ) )
Theorem	acexmidlemv 5530*	Lemma for acexmid 5531. This is acexmid 5531 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 )
Theorem	acexmid 5531*	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 non-empty 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
Syntax	co 5532	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 )
Syntax	coprab 5533	Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class { <. <. x , y >. , z >. | ph }
Syntax	cmpt2 5534	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 )
Definition	df-ov 5535	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 5561 and ovprc2 5562. 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 5536. (Contributed by NM, 28-Feb-1995.)
|- ( A F B ) = ( F ` <. A , B >. )
Definition	df-oprab 5536*	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 5535 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 5656. (Contributed by NM, 12-Mar-1995.)
|- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
Definition	df-mpt2 5537*	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 3841 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 ) }
Theorem	oveq 5538	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( F = G -> ( A F B ) = ( A G B ) )
Theorem	oveq1 5539	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( A F C ) = ( B F C ) )
Theorem	oveq2 5540	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( C F A ) = ( C F B ) )
Theorem	oveq12 5541	Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
|- ( ( A = B /\ C = D ) -> ( A F C ) = ( B F D ) )
Theorem	oveq1i 5542	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( A F C ) = ( B F C )
Theorem	oveq2i 5543	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( C F A ) = ( C F B )
Theorem	oveq12i 5544	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 )
Theorem	oveqi 5545	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
|- A = B   =>    |- ( C A D ) = ( C B D )
Theorem	oveq123i 5546	Equality inference for operation value. (Contributed by FL, 11-Jul-2010.)
|- A = C   &    |- B = D   &    |- F = G   =>    |- ( A F B ) = ( C G D )
Theorem	oveq1d 5547	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A F C ) = ( B F C ) )
Theorem	oveq2d 5548	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C F A ) = ( C F B ) )
Theorem	oveqd 5549	Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D ) = ( C B D ) )
Theorem	oveq12d 5550	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 ) )
Theorem	oveqan12d 5551	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 ) )
Theorem	oveqan12rd 5552	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 ) )
Theorem	oveq123d 5553	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 ) )
Theorem	nfovd 5554	Deduction version of bound-variable hypothesis builder nfov 5555. (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 ) )
Theorem	nfov 5555	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 )
Theorem	oprabidlem 5556*	Slight elaboration of exdistrfor 1721. A lemma for oprabid 5557. (Contributed by Jim Kingdon, 15-Jan-2019.)
|- ( E. x E. y ( x = z /\ ps ) -> E. x ( x = z /\ E. y ps ) )
Theorem	oprabid 5557	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 1439 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
|- ( <. <. x , y >. , z >. e. { <. <. x , y >. , z >. | ph } <-> ph )
Theorem	fnovex 5558	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 )
Theorem	ovexg 5559	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 )
Theorem	ovprc 5560	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 ) = (/) )
Theorem	ovprc1 5561	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 ) = (/) )
Theorem	ovprc2 5562	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 ) = (/) )
Theorem	csbov123g 5563	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 ) )
Theorem	csbov12g 5564*	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 ) )
Theorem	csbov1g 5565*	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 ) )
Theorem	csbov2g 5566*	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 ) )
Theorem	rspceov 5567*	A frequently used special case of rspc2ev 2715 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 ) )
Theorem	fnotovb 5568	Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5236. (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 ) )
Theorem	opabbrex 5569*	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 )
Theorem	0neqopab 5570	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 }
Theorem	brabvv 5571*	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 ) )
Theorem	dfoprab2 5572*	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 ) }
Theorem	reloprab 5573*	An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
|- Rel { <. <. x , y >. , z >. | ph }
Theorem	nfoprab1 5574	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 }
Theorem	nfoprab2 5575	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 }
Theorem	nfoprab3 5576	The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.)
|- F/_ z { <. <. x , y >. , z >. | ph }
Theorem	nfoprab 5577*	Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.)
|- F/ w ph   =>    |- F/_ w { <. <. x , y >. , z >. | ph }
Theorem	oprabbid 5578*	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 } )
Theorem	oprabbidv 5579*	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 } )
Theorem	oprabbii 5580*	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 }
Theorem	ssoprab2 5581	Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4030. (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 } )
Theorem	ssoprab2b 5582	Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4031. (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 ) )
Theorem	eqoprab2b 5583	Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4034. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph <-> ps ) )
Theorem	mpt2eq123 5584*	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 ) )
Theorem	mpt2eq12 5585*	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 ) )
Theorem	mpt2eq123dva 5586*	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 ) )
Theorem	mpt2eq123dv 5587*	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 ) )
Theorem	mpt2eq123i 5588	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 )
Theorem	mpt2eq3dva 5589*	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 ) )
Theorem	mpt2eq3ia 5590	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 )
Theorem	nfmpt21 5591	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 )
Theorem	nfmpt22 5592	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 )
Theorem	nfmpt2 5593*	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 )
Theorem	mpt20 5594	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 ) = (/)
Theorem	oprab4 5595*	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 ) }
Theorem	cbvoprab1 5596*	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 }
Theorem	cbvoprab2 5597*	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 }
Theorem	cbvoprab12 5598*	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 }
Theorem	cbvoprab12v 5599*	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 }
Theorem	cbvoprab3 5600*	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 }