P. 65 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opabex2 6401*	Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ( ph /\ ps ) -> x e. A )   &    |- ( ( ph /\ ps ) -> y e. B )   =>    |- ( ph -> { <. x , y >. | ps } e. _V )
Theorem	opabn1stprc 6402*	An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.)
|- ( E. y ph -> { <. x , y >. | ph } e/ _V )
Theorem	dfxp3 6403*	Define the cross product of three classes. Compare df-xp 4760. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
|- ( ( A X. B ) X. C ) = { <. <. x , y >. , z >. | ( x e. A /\ y e. B /\ z e. C ) }
Theorem	elopabi 6404*	A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
|- ( x = ( 1st ` A ) -> ( ph <-> ps ) )   &    |- ( y = ( 2nd ` A ) -> ( ps <-> ch ) )   =>    |- ( A e. { <. x , y >. | ph } -> ch )
Theorem	eloprabi 6405*	A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)
|- ( x = ( 1st ` ( 1st ` A ) ) -> ( ph <-> ps ) )   &    |- ( y = ( 2nd ` ( 1st ` A ) ) -> ( ps <-> ch ) )   &    |- ( z = ( 2nd ` A ) -> ( ch <-> th ) )   =>    |- ( A e. { <. <. x , y >. , z >. | ph } -> th )
Theorem	mpomptsx 6406*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( x e. A , y e. B |-> C ) = ( z e. U_ x e. A ( { x } X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
Theorem	mpompts 6407*	Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
|- ( x e. A , y e. B |-> C ) = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )
Theorem	dmmpossx 6408*	The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- dom F C_ U_ x e. A ( { x } X. B )
Theorem	fmpox 6409*	Functionality, domain and codomain of a class given by the maps-to notation, where B ( x ) is not constant but depends on x. (Contributed by NM, 29-Dec-2014.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( A. x e. A A. y e. B C e. D <-> F : U_ x e. A ( { x } X. B ) --> D )
Theorem	fmpo 6410*	Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( A. x e. A A. y e. B C e. D <-> F : ( A X. B ) --> D )
Theorem	fnmpo 6411*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( A. x e. A A. y e. B C e. V -> F Fn ( A X. B ) )
Theorem	fnmpoi 6412*	Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
|- F = ( x e. A , y e. B |-> C )   &    |- C e. _V   =>    |- F Fn ( A X. B )
Theorem	dmmpo 6413*	Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
|- F = ( x e. A , y e. B |-> C )   &    |- C e. _V   =>    |- dom F = ( A X. B )
Theorem	mpofvex 6414*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( ( A. x A. y C e. V /\ R e. W /\ S e. X ) -> ( R F S ) e. _V )
Theorem	mpofvexi 6415*	Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- F = ( x e. A , y e. B |-> C )   &    |- C e. _V   &    |- R e. _V   &    |- S e. _V   =>    |- ( R F S ) e. _V
Theorem	ovmpoelrn 6416*	An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)
|- O = ( x e. A , y e. B |-> C )   =>    |- ( ( A. x e. A A. y e. B C e. M /\ X e. A /\ Y e. B ) -> ( X O Y ) e. M )
Theorem	dmmpoga 6417*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6413. (Contributed by Alexander van der Vekens, 10-Feb-2019.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( A. x e. A A. y e. B C e. V -> dom F = ( A X. B ) )
Theorem	dmmpog 6418*	Domain of an operation given by the maps-to notation, closed form of dmmpo 6413. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( C e. V -> dom F = ( A X. B ) )
Theorem	mpoexxg 6419*	Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( ( A e. R /\ A. x e. A B e. S ) -> F e. _V )
Theorem	mpoexg 6420*	Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( ( A e. R /\ B e. S ) -> F e. _V )
Theorem	mpoexga 6421*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.)
|- ( ( A e. V /\ B e. W ) -> ( x e. A , y e. B |-> C ) e. _V )
Theorem	mpoexw 6422*	Weak version of mpoex 6423 that holds without ax-coll 4230. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
|- A e. _V   &    |- B e. _V   &    |- D e. _V   &    |- A. x e. A A. y e. B C e. D   =>    |- ( x e. A , y e. B |-> C ) e. _V
Theorem	mpoex 6423*	If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)
|- A e. _V   &    |- B e. _V   =>    |- ( x e. A , y e. B |-> C ) e. _V
Theorem	fnmpoovd 6424*	A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)
|- ( ph -> M Fn ( A X. B ) )   &    |- ( ( i = a /\ j = b ) -> D = C )   &    |- ( ( ph /\ i e. A /\ j e. B ) -> D e. U )   &    |- ( ( ph /\ a e. A /\ b e. B ) -> C e. V )   =>    |- ( ph -> ( M = ( a e. A , b e. B |-> C ) <-> A. i e. A A. j e. B ( i M j ) = D ) )
Theorem	fmpoco 6425*	Composition of two functions. Variation of fmptco 5848 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> R e. C )   &    |- ( ph -> F = ( x e. A , y e. B |-> R ) )   &    |- ( ph -> G = ( z e. C |-> S ) )   &    |- ( z = R -> S = T )   =>    |- ( ph -> ( G o. F ) = ( x e. A , y e. B |-> T ) )
Theorem	oprabco 6426*	Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
|- ( ( x e. A /\ y e. B ) -> C e. D )   &    |- F = ( x e. A , y e. B |-> C )   &    |- G = ( x e. A , y e. B |-> ( H ` C ) )   =>    |- ( H Fn D -> G = ( H o. F ) )
Theorem	oprab2co 6427*	Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
|- ( ( x e. A /\ y e. B ) -> C e. R )   &    |- ( ( x e. A /\ y e. B ) -> D e. S )   &    |- F = ( x e. A , y e. B |-> <. C , D >. )   &    |- G = ( x e. A , y e. B |-> ( C M D ) )   =>    |- ( M Fn ( R X. S ) -> G = ( M o. F ) )
Theorem	df1st2 6428*	An alternate possible definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- { <. <. x , y >. , z >. | z = x } = ( 1st |` ( _V X. _V ) )
Theorem	df2nd2 6429*	An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )
Theorem	1stconst 6430	The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
|- ( B e. V -> ( 1st |` ( A X. { B } ) ) : ( A X. { B } ) -1-1-onto-> A )
Theorem	2ndconst 6431	The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
|- ( A e. V -> ( 2nd |` ( { A } X. B ) ) : ( { A } X. B ) -1-1-onto-> B )
Theorem	dfmpo 6432*	Alternate definition for the maps-to notation df-mpo 6063 (although it requires that C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
|- C e. _V   =>    |- ( x e. A , y e. B |-> C ) = U_ x e. A U_ y e. B { <. <. x , y >. , C >. }
Theorem	cnvf1olem 6433	Lemma for cnvf1o 6434. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )
Theorem	cnvf1o 6434*	Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( Rel A -> ( x e. A |-> U. `' { x } ) : A -1-1-onto-> `' A )
Theorem	f2ndf 6435	The 2nd (second component of an ordered pair) function restricted to a function F is a function from F into the codomain of F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
|- ( F : A --> B -> ( 2nd |` F ) : F --> B )
Theorem	fo2ndf 6436	The 2nd (second component of an ordered pair) function restricted to a function F is a function from F onto the range of F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
|- ( F : A --> B -> ( 2nd |` F ) : F -onto-> ran F )
Theorem	f1o2ndf1 6437	The 2nd (second component of an ordered pair) function restricted to a one-to-one function F is a one-to-one function from F onto the range of F. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
|- ( F : A -1-1-> B -> ( 2nd |` F ) : F -1-1-onto-> ran F )
Theorem	algrflem 6438	Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- B e. _V   &    |- C e. _V   =>    |- ( B ( F o. 1st ) C ) = ( F ` B )
Theorem	algrflemg 6439	Lemma for algrf 12767 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.)
|- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )
Theorem	xporderlem 6440*	Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
|- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) }   =>    |- ( <. a , b >. T <. c , d >. <-> ( ( ( a e. A /\ c e. A ) /\ ( b e. B /\ d e. B ) ) /\ ( a R c \/ ( a = c /\ b S d ) ) ) )
Theorem	poxp 6441*	A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
|- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) }   =>    |- ( ( R Po A /\ S Po B ) -> T Po ( A X. B ) )
Theorem	spc2ed 6442*	Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)
|- F/ x ch   &    |- F/ y ch   &    |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( ps <-> ch ) )   =>    |- ( ( ph /\ ( A e. V /\ B e. W ) ) -> ( ch -> E. x E. y ps ) )
Theorem	cnvoprab 6443*	The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
|- F/ x ps   &    |- F/ y ps   &    |- ( a = <. x , y >. -> ( ps <-> ph ) )   &    |- ( ps -> a e. ( _V X. _V ) )   =>    |- `' { <. <. x , y >. , z >. | ph } = { <. z , a >. | ps }
Theorem	f1od2 6444*	Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
|- F = ( x e. A , y e. B |-> C )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C e. W )   &    |- ( ( ph /\ z e. D ) -> ( I e. X /\ J e. Y ) )   &    |- ( ph -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( z e. D /\ ( x = I /\ y = J ) ) ) )   =>    |- ( ph -> F : ( A X. B ) -1-1-onto-> D )
Theorem	disjxp1 6445*	The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> Disj x e. A B )   =>    |- ( ph -> Disj x e. A ( B X. C ) )
Theorem	disjsnxp 6446*	The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- Disj j e. A ( { j } X. B )
Theorem	elmpom 6447*	If a maps-to operation is inhabited, the first class it is defined with is inhabited. (Contributed by Jim Kingdon, 4-Mar-2026.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( D e. F -> E. z z e. A )
2.6.16  The support of functions In this section, the support of functions is defined and corresponding theorems are provided. Since basic properties (see suppval 6450) are based on the Axiom of Union (usage of dmexg 5026), these definition and theorems cannot be provided earlier. Until April 2019, the support of a function was represented by the expression ( `' R " ( _V \ { Z } ) ) (see suppimacnvfn 6459). The theorems which are based on this representation and which are provided in previous sections could be moved into this section to have all related theorems in one section, although they do not depend on the Axiom of Union. This was possible because they are not used before. The current theorems differ from the original ones by requiring that the classes representing the "function" (or its "domain") and the "zero element" are sets. Actually, this does not cause any problem (until now).
Syntax	csupp 6448	Extend class definition to include the support of functions.
class supp
Definition	df-supp 6449*	Define the support of a function against a "zero" value. The support of a function is the subset of its domain which is mapped to a value which is not equal to a designed value called the zero value. Note that this definition uses not equal rather than being in terms of an apartness relation (df-ap 8873 or any other apartness relation), and thus is sometimes called "support" rather than "strong support". It is therefore probably most useful when the function has a codomain which has decidable equality and contains the zero value. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
|- supp = ( x e. _V , z e. _V |-> { i e. dom x | ( x " { i } ) =/= { z } } )
Theorem	suppval 6450*	The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
|- ( ( X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )
Theorem	supp0 6451	The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
|- ( Z e. W -> ( (/) supp Z ) = (/) )
Theorem	suppval1 6452*	The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
|- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X ` i ) =/= Z } )
Theorem	suppvalfng 6453*	The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 6454 assumes F is a set rather than its domain X, avoiding ax-coll 4230. (Contributed by SN, 5-Aug-2024.)
|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } )
Theorem	suppvalfn 6454*	The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
|- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } )
Theorem	elsuppfng 6455	An element of the support of a function with a given domain. This version of elsuppfn 6456 assumes F is a set rather than its domain X, avoiding ax-coll 4230. (Contributed by SN, 5-Aug-2024.)
|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( S e. ( F supp Z ) <-> ( S e. X /\ ( F ` S ) =/= Z ) ) )
Theorem	elsuppfn 6456	An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
|- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( S e. ( F supp Z ) <-> ( S e. X /\ ( F ` S ) =/= Z ) ) )
Theorem	fvdifsuppst 6457*	Function value is zero outside of its support. (Contributed by Thierry Arnoux, 21-Jan-2024.)
|- ( ph -> F : A --> B )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. x e. B A. y e. B STAB x = y )   &    |- ( ph -> Z e. B )   &    |- ( ph -> X e. ( A \ ( F supp Z ) ) )   =>    |- ( ph -> ( F ` X ) = Z )
Theorem	cnvimadfsn 6458*	The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
|- ( `' R " ( _V \ { Z } ) ) = { x | E. y ( x R y /\ y =/= Z ) }
Theorem	suppimacnvfn 6459	Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
Theorem	fsuppeq 6460	Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
|- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )
Theorem	fsuppeqg 6461	Version of fsuppeq 6460 avoiding ax-coll 4230 by assuming F is a set rather than its domain I. (Contributed by SN, 30-Jul-2024.)
|- ( ( F e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )
Theorem	suppssdmg 6462	The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.)
|- ( ( F e. V /\ Z e. W ) -> ( F supp Z ) C_ dom F )
Theorem	suppsnopdc 6463	The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
|- F = { <. X , Y >. }   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. W )   &    |- ( ph -> Z e. U )   &    |- ( ph -> DECID Y = Z )   =>    |- ( ph -> ( F supp Z ) = if ( Y = Z , (/) , { X } ) )
Theorem	fvn0elsupp 6464	If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)
|- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> X e. ( G supp (/) ) )
Theorem	fvn0elsuppb 6465	The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)
|- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( ( G ` X ) =/= (/) <-> X e. ( G supp (/) ) ) )
Theorem	rexsupp 6466*	Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)
|- ( ( F Fn X /\ X e. V /\ Z e. W ) -> ( E. x e. ( F supp Z ) ph <-> E. x e. X ( ( F ` x ) =/= Z /\ ph ) ) )
Theorem	ressuppss 6467	The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
|- ( ( F e. V /\ Z e. W ) -> ( ( F |` B ) supp Z ) C_ ( F supp Z ) )
Theorem	mptsuppdifd 6468*	The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
|- F = ( x e. A |-> B )   &    |- ( ph -> A e. V )   &    |- ( ph -> Z e. W )   =>    |- ( ph -> ( F supp Z ) = { x e. A | B e. ( _V \ { Z } ) } )
Theorem	mptsuppd 6469*	The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
|- F = ( x e. A |-> B )   &    |- ( ph -> A e. V )   &    |- ( ph -> Z e. W )   &    |- ( ( ph /\ x e. A ) -> B e. U )   =>    |- ( ph -> ( F supp Z ) = { x e. A | B =/= Z } )
Theorem	suppfnss 6470*	The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 6-Jun-2022.)
|- ( ( ( F Fn A /\ G Fn B ) /\ ( A C_ B /\ B e. V /\ Z e. W ) ) -> ( A. x e. A ( ( G ` x ) = Z -> ( F ` x ) = Z ) -> ( F supp Z ) C_ ( G supp Z ) ) )
Theorem	funsssuppss 6471	The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
|- ( ( Fun G /\ F C_ G /\ G e. V ) -> ( F supp Z ) C_ ( G supp Z ) )
Theorem	fczsupp0 6472	The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
|- ( ( B X. { Z } ) supp Z ) = (/)
Theorem	suppssdc 6473*	Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.)
|- ( ph -> F : A --> B )   &    |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )   &    |- ( ph -> A. x e. A DECID x e. W )   =>    |- ( ph -> ( F supp Z ) C_ W )
Theorem	suppssrst 6474*	A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( F supp Z ) C_ W )   &    |- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. u e. B A. v e. B STAB u = v )   =>    |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )
Theorem	suppssrgst 6475*	A function is zero outside its support. Version of suppssrst 6474 avoiding ax-coll 4230 by assuming F is a set rather than its domain A. (Contributed by SN, 5-May-2024.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( F supp Z ) C_ W )   &    |- ( ph -> F e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. u e. B A. v e. B STAB u = v )   =>    |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )
Theorem	suppssfvg 6476*	Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
|- ( ph -> ( ( x e. D |-> A ) supp Y ) C_ L )   &    |- ( ph -> ( F ` Y ) = Z )   &    |- ( ( ph /\ x e. D ) -> A e. V )   &    |- ( ph -> Y e. U )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( ( x e. D |-> ( F ` A ) ) supp Z ) C_ L )
Theorem	suppofss1dcl 6477*	Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
|- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> G : A --> B )   &    |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u X v ) e. B )   &    |- ( ( ph /\ x e. B ) -> ( Z X x ) = Z )   =>    |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) )
Theorem	suppofss2dcl 6478*	Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
|- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> G : A --> B )   &    |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u X v ) e. B )   &    |- ( ( ph /\ x e. B ) -> ( x X Z ) = Z )   =>    |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) )
Theorem	suppcofn 6479	The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) (Revised by SN, 15-Sep-2023.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
Theorem	supp0cosupp0fn 6480	The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F supp Z ) = (/) -> ( ( F o. G ) supp Z ) = (/) ) )
Theorem	imacosuppfn 6481	The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) )
2.6.17  Special maps-to operations The following theorems are about maps-to operations (see df-mpo 6063) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 6139, ovmpox 6190 and fmpox 6409). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short.
Theorem	opeliunxp2f 6482*	Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4900. (Contributed by AV, 25-Oct-2020.)
|- F/_ x E   &    |- ( x = C -> B = E )   =>    |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
Theorem	mpoxopn0yelv 6483*	If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> C )   =>    |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) -> K e. V ) )
Theorem	mpoxopoveq 6484*	Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )   =>    |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )
Theorem	mpoxopovel 6485*	Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )   =>    |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) <-> ( K e. V /\ N e. V /\ [. <. V , W >. / x ]. [. K / y ]. [. N / n ]. ph ) ) )
Theorem	rbropapd 6486*	Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
|- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )   &    |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )
Theorem	rbropap 6487*	Properties of a pair in a restricted binary relation M expressed as an ordered-pair class abstraction: M is the binary relation W restricted by the condition ps. (Contributed by AV, 31-Jan-2021.)
|- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )   &    |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )   =>    |- ( ( ph /\ F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) )
2.6.18  Function transposition
Syntax	ctpos 6488	The transposition of a function.
class tpos F
Definition	df-tpos 6489*	Define the transposition of a function, which is a function G = tpos F satisfying G ( x , y ) = F ( y , x ). (Contributed by Mario Carneiro, 10-Sep-2015.)
|- tpos F = ( F o. ( x e. ( `' dom F u. { (/) } ) |-> U. `' { x } ) )
Theorem	tposss 6490	Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F C_ G -> tpos F C_ tpos G )
Theorem	tposeq 6491	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F = G -> tpos F = tpos G )
Theorem	tposeqd 6492	Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> F = G )   =>    |- ( ph -> tpos F = tpos G )
Theorem	tposssxp 6493	The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
|- tpos F C_ ( ( `' dom F u. { (/) } ) X. ran F )
Theorem	reltpos 6494	The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- Rel tpos F
Theorem	brtpos2 6495	Value of the transposition at a pair <. A , B >.. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( B e. V -> ( Atpos F B <-> ( A e. ( `' dom F u. { (/) } ) /\ U. `' { A } F B ) ) )
Theorem	brtpos0 6496	The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( A e. V -> ( (/)tpos F A <-> (/) F A ) )
Theorem	reldmtpos 6497	Necessary and sufficient condition for dom tpos F to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom tpos F <-> -. (/) e. dom F )
Theorem	brtposg 6498	The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. A , B >.tpos F C <-> <. B , A >. F C ) )
Theorem	ottposg 6499	The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. A , B , C >. e. tpos F <-> <. B , A , C >. e. F ) )
Theorem	dmtpos 6500	The domain of tpos F when dom F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> dom tpos F = `' dom F )