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	fo2ndf 6401	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 6402	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 6403	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 6404	Lemma for algrf 12680 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 6405*	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 6406*	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 6407*	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 6408*	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 6409*	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 6410*	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 6411*	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 6412*	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 6415) are based on the Axiom of Union (usage of dmexg 5002), 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 6424). 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 6413	Extend class definition to include the support of functions.
class supp
Definition	df-supp 6414*	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 8804 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 6415*	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 6416	The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
|- ( Z e. W -> ( (/) supp Z ) = (/) )
Theorem	suppval1 6417*	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 6418*	The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 6419 assumes F is a set rather than its domain X, avoiding ax-coll 4209. (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 6419*	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 6420	An element of the support of a function with a given domain. This version of elsuppfn 6421 assumes F is a set rather than its domain X, avoiding ax-coll 4209. (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 6421	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 6422*	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 6423*	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 6424	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 6425	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 6426	Version of fsuppeq 6425 avoiding ax-coll 4209 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 6427	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 6428	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 6429	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 6430	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 6431*	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 6432	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 6433*	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 6434*	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 6435*	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 6436	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 6437	The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
|- ( ( B X. { Z } ) supp Z ) = (/)
Theorem	suppssdc 6438*	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 6439*	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 6440*	A function is zero outside its support. Version of suppssrst 6439 avoiding ax-coll 4209 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 6441*	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 6442*	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 6443*	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 6444	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 6445	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 6446	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 6033) 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 6109, ovmpox 6160 and fmpox 6374). 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 6447*	Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4876. (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 6448*	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 6449*	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 6450*	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 6451*	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 6452*	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 6453	The transposition of a function.
class tpos F
Definition	df-tpos 6454*	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 6455	Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F C_ G -> tpos F C_ tpos G )
Theorem	tposeq 6456	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F = G -> tpos F = tpos G )
Theorem	tposeqd 6457	Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> F = G )   =>    |- ( ph -> tpos F = tpos G )
Theorem	tposssxp 6458	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 6459	The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- Rel tpos F
Theorem	brtpos2 6460	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 6461	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 6462	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 6463	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 6464	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 6465	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 )
Theorem	rntpos 6466	The range of tpos F when dom F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> ran tpos F = ran F )
Theorem	tposexg 6467	The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F e. V -> tpos F e. _V )
Theorem	ovtposg 6468	The transposition swaps the arguments in a two-argument function. When F is a matrix, which is to say a function from ( 1 ... m ) X. ( 1 ... n ) to the reals or some ring, tpos F is the transposition of F, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )
Theorem	tposfun 6469	The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Fun F -> Fun tpos F )
Theorem	dftpos2 6470*	Alternate definition of tpos when F has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> tpos F = ( F o. ( x e. `' dom F |-> U. `' { x } ) ) )
Theorem	dftpos3 6471*	Alternate definition of tpos when F has relational domain. Compare df-cnv 4739. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> tpos F = { <. <. x , y >. , z >. | <. y , x >. F z } )
Theorem	dftpos4 6472*	Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- tpos F = ( F o. ( x e. ( ( _V X. _V ) u. { (/) } ) |-> U. `' { x } ) )
Theorem	tpostpos 6473	Value of the double transposition for a general class F. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- tpos tpos F = ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) )
Theorem	tpostpos2 6474	Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- ( ( Rel F /\ Rel dom F ) -> tpos tpos F = F )
Theorem	tposfn2 6475	The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )
Theorem	tposfo2 6476	Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A -onto-> B -> tpos F : `' A -onto-> B ) )
Theorem	tposf2 6477	The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) )
Theorem	tposf12 6478	Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A -1-1-> B -> tpos F : `' A -1-1-> B ) )
Theorem	tposf1o2 6479	Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A -1-1-onto-> B -> tpos F : `' A -1-1-onto-> B ) )
Theorem	tposfo 6480	The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( F : ( A X. B ) -onto-> C -> tpos F : ( B X. A ) -onto-> C )
Theorem	tposf 6481	The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( F : ( A X. B ) --> C -> tpos F : ( B X. A ) --> C )
Theorem	tposfn 6482	Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- ( F Fn ( A X. B ) -> tpos F Fn ( B X. A ) )
Theorem	tpos0 6483	Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
|- tpos (/) = (/)
Theorem	tposco 6484	Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- tpos ( F o. G ) = ( F o. tpos G )
Theorem	tpossym 6485*	Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- ( F Fn ( A X. A ) -> (tpos F = F <-> A. x e. A A. y e. A ( x F y ) = ( y F x ) ) )
Theorem	tposeqi 6486	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F = G   =>    |- tpos F = tpos G
Theorem	tposex 6487	A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F e. _V   =>    |- tpos F e. _V
Theorem	nftpos 6488	Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F/_ x F   =>    |- F/_ xtpos F
Theorem	tposoprab 6489*	Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F = { <. <. x , y >. , z >. | ph }   =>    |- tpos F = { <. <. y , x >. , z >. | ph }
Theorem	tposmpo 6490*	Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- tpos F = ( y e. B , x e. A |-> C )
2.6.19  Undefined values
Theorem	pwuninel2 6491	The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( U. A e. V -> -. ~P U. A e. A )
Theorem	2pwuninelg 6492	The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.)
|- ( A e. V -> -. ~P ~P U. A e. A )
2.6.20  Functions on ordinals; strictly monotone ordinal functions
Theorem	iunon 6493*	The indexed union of a set of ordinal numbers B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
|- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On )
Syntax	wsmo 6494	Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
wff Smo A
Definition	df-smo 6495*	Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
|- ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) )
Theorem	dfsmo2 6496*	Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
|- ( Smo F <-> ( F : dom F --> On /\ Ord dom F /\ A. x e. dom F A. y e. x ( F ` y ) e. ( F ` x ) ) )
Theorem	issmo 6497*	Conditions for which A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
|- A : B --> On   &    |- Ord B   &    |- ( ( x e. B /\ y e. B ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )   &    |- dom A = B   =>    |- Smo A
Theorem	issmo2 6498*	Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
|- ( F : A --> B -> ( ( B C_ On /\ Ord A /\ A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) ) -> Smo F ) )
Theorem	smoeq 6499	Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- ( A = B -> ( Smo A <-> Smo B ) )
Theorem	smodm 6500	The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- ( Smo A -> Ord dom A )