P. 60 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5901-6000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cbvriotav 5901*	Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )
Theorem	csbriotag 5902*	Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
|- ( A e. V -> [_ A / x ]_ ( iota_ y e. B ph ) = ( iota_ y e. B [. A / x ]. ph ) )
Theorem	riotacl2 5903	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 5904*	Closure of restricted iota. (Contributed by NM, 21-Aug-2011.)
|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
Theorem	riotasbc 5905	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 5906*	Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2759 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 5907	Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2756 analog.) (Contributed by NM, 16-Jan-2012.)
|- ( x e. A -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ x e. A ps )
Theorem	riota1 5908*	Property of restricted iota. Compare iota1 5243. (Contributed by Mario Carneiro, 15-Oct-2016.)
|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )
Theorem	riota1a 5909	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 5910*	A deduction version of riota2f 5911. (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 5911*	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 5912*	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 5913*	Properties of a restricted definite description operator. Todo (df-riota 5889 update): can some uses of riota2f 5911 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 5914*	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 5915*	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 5916*	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 5917*	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 5918*	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 5919	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 5920*	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 5921*	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 5922*	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 5923*	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 5924*	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 5925*	Lemma for acexmid 5933. (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 5926*	Lemma for acexmid 5933. (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 5927*	Lemma for acexmid 5933. (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 5928*	Lemma for acexmid 5933. (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 5929*	Lemma for acexmid 5933. 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 5270. 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 5930*	Lemma for acexmid 5933. List the cases identified in acexmidlemcase 5929 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 5931*	Lemma for acexmid 5933. This builds on acexmidlem1 5930 by noting that every element of C is inhabited. (Note that y is not quite a function in the df-fun 5270 sense because it uses ordered pairs as described in opthreg 4602 rather than df-op 3641). The set A is also found in onsucelsucexmidlem 4575. (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 5932*	Lemma for acexmid 5933. This is acexmid 5933 with additional disjoint variable conditions, 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 5933*	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). For this theorem stated using the df-ac 7300 and df-exmid 4238 syntaxes, see exmidac 7303. (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 )
2.6.11  Operations
Syntax	co 5934	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 5935	Extend class notation to include class abstraction (class builder) of nested ordered pairs.
class { <. <. x , y >. , z >. | ph }
Syntax	cmpo 5936	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 5937	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 5971 and ovprc2 5972. 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 5938. (Contributed by NM, 28-Feb-1995.)
|- ( A F B ) = ( F ` <. A , B >. )
Definition	df-oprab 5938*	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 5937 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 ovmpo 6071. (Contributed by NM, 12-Mar-1995.)
|- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
Definition	df-mpo 5939*	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 4106 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 5940	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( F = G -> ( A F B ) = ( A G B ) )
Theorem	oveq1 5941	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( A F C ) = ( B F C ) )
Theorem	oveq2 5942	Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.)
|- ( A = B -> ( C F A ) = ( C F B ) )
Theorem	oveq12 5943	Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.)
|- ( ( A = B /\ C = D ) -> ( A F C ) = ( B F D ) )
Theorem	oveq1i 5944	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( A F C ) = ( B F C )
Theorem	oveq2i 5945	Equality inference for operation value. (Contributed by NM, 28-Feb-1995.)
|- A = B   =>    |- ( C F A ) = ( C F B )
Theorem	oveq12i 5946	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 5947	Equality inference for operation value. (Contributed by NM, 24-Nov-2007.)
|- A = B   =>    |- ( C A D ) = ( C B D )
Theorem	oveq123i 5948	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 5949	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A F C ) = ( B F C ) )
Theorem	oveq2d 5950	Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C F A ) = ( C F B ) )
Theorem	oveqd 5951	Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D ) = ( C B D ) )
Theorem	oveq12d 5952	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 5953	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 5954	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 5955	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	fvoveq1d 5956	Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )
Theorem	fvoveq1 5957	Equality theorem for nested function and operation value. Closed form of fvoveq1d 5956. (Contributed by AV, 23-Jul-2022.)
|- ( A = B -> ( F ` ( A O C ) ) = ( F ` ( B O C ) ) )
Theorem	ovanraleqv 5958*	Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
|- ( B = X -> ( ph <-> ps ) )   =>    |- ( B = X -> ( A. x e. V ( ph /\ ( A .x. B ) = C ) <-> A. x e. V ( ps /\ ( A .x. X ) = C ) ) )
Theorem	imbrov2fvoveq 5959	Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)
|- ( X = Y -> ( ph <-> ps ) )   =>    |- ( X = Y -> ( ( ph -> ( F ` ( ( G ` X ) .x. O ) ) R A ) <-> ( ps -> ( F ` ( ( G ` Y ) .x. O ) ) R A ) ) )
Theorem	ovrspc2v 5960*	If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.)
|- ( ( ( X e. A /\ Y e. B ) /\ A. x e. A A. y e. B ( x F y ) e. C ) -> ( X F Y ) e. C )
Theorem	oveqrspc2v 5961*	Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.)
|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x F y ) = ( x G y ) )   =>    |- ( ( ph /\ ( X e. A /\ Y e. B ) ) -> ( X F Y ) = ( X G Y ) )
Theorem	oveqdr 5962	Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.)
|- ( ph -> F = G )   =>    |- ( ( ph /\ ps ) -> ( x F y ) = ( x G y ) )
Theorem	nfovd 5963	Deduction version of bound-variable hypothesis builder nfov 5964. (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 5964	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 5965*	Slight elaboration of exdistrfor 1822. A lemma for oprabid 5966. (Contributed by Jim Kingdon, 15-Jan-2019.)
|- ( E. x E. y ( x = z /\ ps ) -> E. x ( x = z /\ E. y ps ) )
Theorem	oprabid 5966	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable condition between x, y, and z, we use ax-bndl 1531 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
|- ( <. <. x , y >. , z >. e. { <. <. x , y >. , z >. | ph } <-> ph )
Theorem	fnovex 5967	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 5968	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	ovssunirng 5969	The result of an operation value is always a subset of the union of the range. (Contributed by Mario Carneiro, 12-Jan-2017.)
|- ( ( X e. V /\ Y e. W ) -> ( X F Y ) C_ U. ran F )
Theorem	ovprc 5970	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 5971	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 5972	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 5973	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 5974*	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 5975*	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 5976*	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 5977*	A frequently used special case of rspc2ev 2891 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 5978	Equivalence of operation value and ordered triple membership, analogous to fnopfvb 5614. (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 5979*	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 5980	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 5981*	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 5982*	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 5983*	An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)
|- Rel { <. <. x , y >. , z >. | ph }
Theorem	nfoprab1 5984	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 5985	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 5986	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 5987*	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 5988*	Equivalent wff's yield equal operation class abstractions (deduction form). (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 5989*	Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )
Theorem	oprabbii 5990*	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 5991	Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4320. (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 5992	Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4321. (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 5993	Equivalence of ordered pair abstraction subclass and biconditional. Compare eqopab2b 4324. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , z >. | ps } <-> A. x A. y A. z ( ph <-> ps ) )
Theorem	mpoeq123 5994*	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	mpoeq12 5995*	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	mpoeq123dva 5996*	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	mpoeq123dv 5997*	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	mpoeq123i 5998	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	mpoeq3dva 5999*	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	mpoeq3ia 6000	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 )