P. 54 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5301-5400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iotabii 5301	Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ph <-> ps )   =>    |- ( iota x ph ) = ( iota x ps )
Theorem	iotacl 5302	Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 5277). (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( E! x ph -> ( iota x ph ) e. { x | ph } )
Theorem	iota2df 5303	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 5304*	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 5305*	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 5306	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 5307*	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 5308	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 5309*	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 5310*	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 5311	Extend the definition of a wff to include the function predicate. (Read: A is a function.)
wff Fun A
Syntax	wfn 5312	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 5313	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 5314	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 5315	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 5316	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 5317	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 5318	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 5319	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 5349). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4144 with the maps-to notation (see df-mpt 4146). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5320), a function with a given domain and codomain (df-f 5321), a one-to-one function (df-f1 5322), an onto function (df-fo 5323), or a one-to-one onto function (df-f1o 5324). For alternate definitions, see dffun2 5327, dffun4 5328, dffun6 5331, dffun7 5344, dffun8 5345, and dffun9 5346. (Contributed by NM, 1-Aug-1994.)
|- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) )
Definition	df-fn 5320	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 5321	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 5322	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 5323	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 5324	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 5325*	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 4146), 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 5326*	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 5327*	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 5328*	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 5329*	A way of proving a relation is a function, analogous to mo2r 2130. (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 5330*	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 5331*	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 5332*	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 5333*	Definition of function like dffun4 5328 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 5334	A function is a relation. (Contributed by NM, 1-Aug-1994.)
|- ( Fun A -> Rel A )
Theorem	0nelfun 5335	A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
|- ( Fun R -> (/) e/ R )
Theorem	funss 5336	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 5337	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
|- ( A = B -> ( Fun A <-> Fun B ) )
Theorem	funeqi 5338	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- A = B   =>    |- ( Fun A <-> Fun B )
Theorem	funeqd 5339	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ( Fun A <-> Fun B ) )
Theorem	nffun 5340	Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
|- F/_ x F   =>    |- F/ x Fun F
Theorem	sbcfung 5341	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 5342*	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 5343*	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 5344*	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 5345 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 5345*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5344. (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 5346*	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 5347	An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
|- ( Fun A <-> A Fn dom A )
Theorem	funfnd 5348	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 5349	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 5350	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
|- -. Fun _V
Theorem	funopg 5351	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 5352*	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 5353*	A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
|- Fun { <. x , y >. | y = A }
Theorem	funopab4 5354*	A class of ordered pairs of values in the form used by df-mpt 4146 is a function. (Contributed by NM, 17-Feb-2013.)
|- Fun { <. x , y >. | ( ph /\ y = A ) }
Theorem	funmpt 5355	A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
|- Fun ( x e. A |-> B )
Theorem	funmpt2 5356	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 5357	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 5358	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 5359	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 5360	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 5361	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	fununmo 5362*	If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)
|- ( Fun ( F u. G ) -> E* y x F y )
Theorem	fununfun 5363	If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)
|- ( Fun ( F u. G ) -> ( Fun F /\ Fun G ) )
Theorem	fundif 5364	A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
|- ( Fun F -> Fun ( F \ A ) )
Theorem	funcnvsn 5365	The converse singleton of an ordered pair is a function. This is equivalent to funsn 5368 via cnvsn 5210, 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 5366	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 5367	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 5368	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 5369	A function based on the singleton of an ordered pair. Unlike funsng 5366, 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 5370	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 5371	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 5372	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 5373	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 5374	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 5375	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 5376	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 5377	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 5378	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
|- Fun (/)
Theorem	funcnvcnv 5379	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
|- ( Fun A -> Fun `' `' A )
Theorem	funcnv2 5380*	A simpler equivalence for single-rooted (see funcnv 5381). (Contributed by NM, 9-Aug-2004.)
|- ( Fun `' A <-> A. y E* x x A y )
Theorem	funcnv 5381*	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 5380 for a simpler version. (Contributed by NM, 13-Aug-2004.)
|- ( Fun `' A <-> A. y e. ran A E* x x A y )
Theorem	funcnv3 5382*	A condition showing a class is single-rooted. (See funcnv 5381). (Contributed by NM, 26-May-2006.)
|- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y )
Theorem	funcnveq 5383*	Another way of expressing that a class is single-rooted. Counterpart to dffun2 5327. (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 5384*	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 5385	A single-valued relation is a function. (See fun2cnv 5384 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 5386*	Single-rootedness (see funcnv 5381) 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 5387*	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 5388*	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 5389*	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5381 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 5390*	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 5391	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 5392	The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
|- ( Fun `' F -> Fun `' ( F |` A ) )
Theorem	funcnvres 5393	The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
|- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )
Theorem	cnvresid 5394	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
|- `' ( _I |` A ) = ( _I |` A )
Theorem	funcnvres2 5395	The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
|- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )
Theorem	funimacnv 5396	The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
|- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )
Theorem	funimass1 5397	A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
|- ( ( Fun F /\ A C_ ran F ) -> ( ( `' F " A ) C_ B -> A C_ ( F " B ) ) )
Theorem	funimass2 5398	A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
|- ( ( Fun F /\ A C_ ( `' F " B ) ) -> ( F " A ) C_ B )
Theorem	imadiflem 5399	One direction of imadif 5400. This direction does not require Fun `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
|- ( ( F " A ) \ ( F " B ) ) C_ ( F " ( A \ B ) )
Theorem	imadif 5400	The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
|- ( Fun `' F -> ( F " ( A \ B ) ) = ( ( F " A ) \ ( F " B ) ) )