P. 53 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5201-5300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iotaval 5201*	Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
Theorem	iotauni 5202	Equivalence between two different forms of iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
Theorem	iotaint 5203	Equivalence between two different forms of iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )
Theorem	iota1 5204	Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
Theorem	iotanul 5205	Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one x that satisfies ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( -. E! x ph -> ( iota x ph ) = (/) )
Theorem	euiotaex 5206	Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the iota class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
|- ( E! x ph -> ( iota x ph ) e. _V )
Theorem	iotass 5207*	Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
|- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )
Theorem	iota4 5208	Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
Theorem	iota4an 5209	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )
Theorem	iota5 5210*	A method for computing iota. (Contributed by NM, 17-Sep-2013.)
|- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) )   =>    |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A )
Theorem	iotabidv 5211*	Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( iota x ps ) = ( iota x ch ) )
Theorem	iotabii 5212	Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ph <-> ps )   =>    |- ( iota x ph ) = ( iota x ps )
Theorem	iotacl 5213	Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 5190). (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( E! x ph -> ( iota x ph ) e. { x | ph } )
Theorem	iota2df 5214	A condition that allows us to represent "the unique element such that ph " with a class expression A. (Contributed by NM, 30-Dec-2014.)
|- ( ph -> B e. V )   &    |- ( ph -> E! x ps )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   &    |- F/ x ph   &    |- ( ph -> F/ x ch )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> ( ch <-> ( iota x ps ) = B ) )
Theorem	iota2d 5215*	A condition that allows us to represent "the unique element such that ph " with a class expression A. (Contributed by NM, 30-Dec-2014.)
|- ( ph -> B e. V )   &    |- ( ph -> E! x ps )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( ch <-> ( iota x ps ) = B ) )
Theorem	eliota 5216*	An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
|- ( A e. ( iota x ph ) <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )
Theorem	eliotaeu 5217	An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
|- ( A e. ( iota x ph ) -> E! x ph )
Theorem	iota2 5218*	The unique element such that ph. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
Theorem	sniota 5219	A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- ( E! x ph -> { x | ph } = { ( iota x ph ) } )
Theorem	iotam 5220*	Representation of "the unique element such that ph " with a class expression A which is inhabited (that means that "the unique element such that ph " exists). (Contributed by AV, 30-Jan-2024.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ E. w w e. A /\ A = ( iota x ph ) ) -> ps )
Theorem	csbiotag 5221*	Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
|- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )
2.6.8  Functions
Syntax	wfun 5222	Extend the definition of a wff to include the function predicate. (Read: A is a function.)
wff Fun A
Syntax	wfn 5223	Extend the definition of a wff to include the function predicate with a domain. (Read: A is a function on B.)
wff A Fn B
Syntax	wf 5224	Extend the definition of a wff to include the function predicate with domain and codomain. (Read: F maps A into B.)
wff F : A --> B
Syntax	wf1 5225	Extend the definition of a wff to include one-to-one functions. (Read: F maps A one-to-one into B.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
wff F : A -1-1-> B
Syntax	wfo 5226	Extend the definition of a wff to include onto functions. (Read: F maps A onto B.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
wff F : A -onto-> B
Syntax	wf1o 5227	Extend the definition of a wff to include one-to-one onto functions. (Read: F maps A one-to-one onto B.) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.
wff F : A -1-1-onto-> B
Syntax	cfv 5228	Extend the definition of a class to include the value of a function. (Read: The value of F at A, or " F of A.")
class ( F ` A )
Syntax	wiso 5229	Extend the definition of a wff to include the isomorphism property. (Read: H is an R, S isomorphism of A onto B.)
wff H Isom R , S ( A , B )
Definition	df-fun 5230	Define predicate that determines if some class A is a function. Definition 10.1 of [Quine] p. 65. For example, the expression Fun _I is true (funi 5260). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4076 with the maps-to notation (see df-mpt 4078). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5231), a function with a given domain and codomain (df-f 5232), a one-to-one function (df-f1 5233), an onto function (df-fo 5234), or a one-to-one onto function (df-f1o 5235). For alternate definitions, see dffun2 5238, dffun4 5239, dffun6 5242, dffun7 5255, dffun8 5256, and dffun9 5257. (Contributed by NM, 1-Aug-1994.)
|- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) )
Definition	df-fn 5231	Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
|- ( A Fn B <-> ( Fun A /\ dom A = B ) )
Definition	df-f 5232	Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
|- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
Definition	df-f1 5233	Define a one-to-one function. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.)
|- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
Definition	df-fo 5234	Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). (Contributed by NM, 1-Aug-1994.)
|- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
Definition	df-f1o 5235	Define a one-to-one onto function. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.)
|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
Definition	df-fv 5236*	Define the value of a function, ( F ` A ), also known as function application. For example, ( _I ` (/) ) = (/). Typically, function F is defined using maps-to notation (see df-mpt 4078), but this is not required. For example, F = { <. 2 , 6 >. , <. 3 , 9 >. } -> ( F ` 3 ) = 9. We will later define two-argument functions using ordered pairs as ( A F B ) = ( F ` <. A , B >. ). This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful. The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar F ( A ) notation for a function's value at A, i.e., " F of A," but without context-dependent notational ambiguity. (Contributed by NM, 1-Aug-1994.) Revised to use iota. (Revised by Scott Fenton, 6-Oct-2017.)
|- ( F ` A ) = ( iota x A F x )
Definition	df-isom 5237*	Define the isomorphism predicate. We read this as " H is an R, S isomorphism of A onto B". Normally, R and S are ordering relations on A and B respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that R and S are subscripts. (Contributed by NM, 4-Mar-1997.)
|- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
Theorem	dffun2 5238*	Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
|- ( Fun A <-> ( Rel A /\ A. x A. y A. z ( ( x A y /\ x A z ) -> y = z ) ) )
Theorem	dffun4 5239*	Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)
|- ( Fun A <-> ( Rel A /\ A. x A. y A. z ( ( <. x , y >. e. A /\ <. x , z >. e. A ) -> y = z ) ) )
Theorem	dffun5r 5240*	A way of proving a relation is a function, analogous to mo2r 2088. (Contributed by Jim Kingdon, 27-May-2020.)
|- ( ( Rel A /\ A. x E. z A. y ( <. x , y >. e. A -> y = z ) ) -> Fun A )
Theorem	dffun6f 5241*	Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   &    |- F/_ y A   =>    |- ( Fun A <-> ( Rel A /\ A. x E* y x A y ) )
Theorem	dffun6 5242*	Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)
|- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) )
Theorem	funmo 5243*	A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
|- ( Fun F -> E* y A F y )
Theorem	dffun4f 5244*	Definition of function like dffun4 5239 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.)
|- F/_ x A   &    |- F/_ y A   &    |- F/_ z A   =>    |- ( Fun A <-> ( Rel A /\ A. x A. y A. z ( ( <. x , y >. e. A /\ <. x , z >. e. A ) -> y = z ) ) )
Theorem	funrel 5245	A function is a relation. (Contributed by NM, 1-Aug-1994.)
|- ( Fun A -> Rel A )
Theorem	0nelfun 5246	A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
|- ( Fun R -> (/) e/ R )
Theorem	funss 5247	Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
|- ( A C_ B -> ( Fun B -> Fun A ) )
Theorem	funeq 5248	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
|- ( A = B -> ( Fun A <-> Fun B ) )
Theorem	funeqi 5249	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- A = B   =>    |- ( Fun A <-> Fun B )
Theorem	funeqd 5250	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ( Fun A <-> Fun B ) )
Theorem	nffun 5251	Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
|- F/_ x F   =>    |- F/ x Fun F
Theorem	sbcfung 5252	Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
|- ( A e. V -> ( [. A / x ]. Fun F <-> Fun [_ A / x ]_ F ) )
Theorem	funeu 5253*	There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( Fun F /\ A F B ) -> E! y A F y )
Theorem	funeu2 5254*	There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
|- ( ( Fun F /\ <. A , B >. e. F ) -> E! y <. A , y >. e. F )
Theorem	dffun7 5255*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun8 5256 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
|- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y x A y ) )
Theorem	dffun8 5256*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5255. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( Fun A <-> ( Rel A /\ A. x e. dom A E! y x A y ) )
Theorem	dffun9 5257*	Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
|- ( Fun A <-> ( Rel A /\ A. x e. dom A E* y e. ran A x A y ) )
Theorem	funfn 5258	An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
|- ( Fun A <-> A Fn dom A )
Theorem	funfnd 5259	A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> Fun A )   =>    |- ( ph -> A Fn dom A )
Theorem	funi 5260	The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
|- Fun _I
Theorem	nfunv 5261	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
|- -. Fun _V
Theorem	funopg 5262	A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B )
Theorem	funopab 5263*	A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)
|- ( Fun { <. x , y >. | ph } <-> A. x E* y ph )
Theorem	funopabeq 5264*	A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
|- Fun { <. x , y >. | y = A }
Theorem	funopab4 5265*	A class of ordered pairs of values in the form used by df-mpt 4078 is a function. (Contributed by NM, 17-Feb-2013.)
|- Fun { <. x , y >. | ( ph /\ y = A ) }
Theorem	funmpt 5266	A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
|- Fun ( x e. A |-> B )
Theorem	funmpt2 5267	Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
|- F = ( x e. A |-> B )   =>    |- Fun F
Theorem	funco 5268	The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) )
Theorem	funres 5269	A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)
|- ( Fun F -> Fun ( F |` A ) )
Theorem	funssres 5270	The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)
|- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
Theorem	fun2ssres 5271	Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
|- ( ( Fun F /\ G C_ F /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )
Theorem	funun 5272	The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( Fun F /\ Fun G ) /\ ( dom F i^i dom G ) = (/) ) -> Fun ( F u. G ) )
Theorem	funcnvsn 5273	The converse singleton of an ordered pair is a function. This is equivalent to funsn 5276 via cnvsn 5123, but stating it this way allows us to skip the sethood assumptions on A and B. (Contributed by NM, 30-Apr-2015.)
|- Fun `' { <. A , B >. }
Theorem	funsng 5274	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
|- ( ( A e. V /\ B e. W ) -> Fun { <. A , B >. } )
Theorem	fnsng 5275	Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> { <. A , B >. } Fn { A } )
Theorem	funsn 5276	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
|- A e. _V   &    |- B e. _V   =>    |- Fun { <. A , B >. }
Theorem	funinsn 5277	A function based on the singleton of an ordered pair. Unlike funsng 5274, this holds even if A or B is a proper class. (Contributed by Jim Kingdon, 17-Apr-2022.)
|- Fun ( { <. A , B >. } i^i ( V X. W ) )
Theorem	funprg 5278	A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> Fun { <. A , C >. , <. B , D >. } )
Theorem	funtpg 5279	A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
|- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Fun { <. X , A >. , <. Y , B >. , <. Z , C >. } )
Theorem	funpr 5280	A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   =>    |- ( A =/= B -> Fun { <. A , C >. , <. B , D >. } )
Theorem	funtp 5281	A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- E e. _V   &    |- F e. _V   =>    |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> Fun { <. A , D >. , <. B , E >. , <. C , F >. } )
Theorem	fnsn 5282	Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- A e. _V   &    |- B e. _V   =>    |- { <. A , B >. } Fn { A }
Theorem	fnprg 5283	Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( ( ( A e. V /\ B e. W ) /\ ( C e. X /\ D e. Y ) /\ A =/= B ) -> { <. A , C >. , <. B , D >. } Fn { A , B } )
Theorem	fntpg 5284	Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
|- ( ( ( X e. U /\ Y e. V /\ Z e. W ) /\ ( A e. F /\ B e. G /\ C e. H ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { <. X , A >. , <. Y , B >. , <. Z , C >. } Fn { X , Y , Z } )
Theorem	fntp 5285	A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- D e. _V   &    |- E e. _V   &    |- F e. _V   =>    |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> { <. A , D >. , <. B , E >. , <. C , F >. } Fn { A , B , C } )
Theorem	fun0 5286	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
|- Fun (/)
Theorem	funcnvcnv 5287	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
|- ( Fun A -> Fun `' `' A )
Theorem	funcnv2 5288*	A simpler equivalence for single-rooted (see funcnv 5289). (Contributed by NM, 9-Aug-2004.)
|- ( Fun `' A <-> A. y E* x x A y )
Theorem	funcnv 5289*	The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that x A y. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5288 for a simpler version. (Contributed by NM, 13-Aug-2004.)
|- ( Fun `' A <-> A. y e. ran A E* x x A y )
Theorem	funcnv3 5290*	A condition showing a class is single-rooted. (See funcnv 5289). (Contributed by NM, 26-May-2006.)
|- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y )
Theorem	funcnveq 5291*	Another way of expressing that a class is single-rooted. Counterpart to dffun2 5238. (Contributed by Jim Kingdon, 24-Dec-2018.)
|- ( Fun `' A <-> A. x A. y A. z ( ( x A y /\ z A y ) -> x = z ) )
Theorem	fun2cnv 5292*	The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that A is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
|- ( Fun `' `' A <-> A. x E* y x A y )
Theorem	svrelfun 5293	A single-valued relation is a function. (See fun2cnv 5292 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
|- ( Fun A <-> ( Rel A /\ Fun `' `' A ) )
Theorem	fncnv 5294*	Single-rootedness (see funcnv 5289) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
|- ( `' ( R i^i ( A X. B ) ) Fn B <-> A. y e. B E! x e. A x R y )
Theorem	fun11 5295*	Two ways of stating that A is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
|- ( ( Fun `' `' A /\ Fun `' A ) <-> A. x A. y A. z A. w ( ( x A y /\ z A w ) -> ( x = z <-> y = w ) ) )
Theorem	fununi 5296*	The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
|- ( A. f e. A ( Fun f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun U. A )
Theorem	funcnvuni 5297*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5289 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
|- ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun `' U. A )
Theorem	fun11uni 5298*	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
|- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( Fun U. A /\ Fun `' U. A ) )
Theorem	funin 5299	The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|- ( Fun F -> Fun ( F i^i G ) )
Theorem	funres11 5300	The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
|- ( Fun `' F -> Fun `' ( F |` A ) )