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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfunss 5001 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 ) )
 
Theoremfuneq 5002 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremfuneqi 5003 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  A  =  B   =>    |-  ( Fun  A  <->  Fun 
 B )
 
Theoremfuneqd 5004 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremnffun 5005 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
 |-  F/_ x F   =>    |- 
 F/ x Fun  F
 
Theoremsbcfung 5006 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 ) )
 
Theoremfuneu 5007* 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 )
 
Theoremfuneu2 5008* 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 )
 
Theoremdffun7 5009* 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 5010 shows that it doesn't 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 ) )
 
Theoremdffun8 5010* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5009. (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 ) )
 
Theoremdffun9 5011* 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 ) )
 
Theoremfunfn 5012 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  A  <->  A  Fn  dom  A )
 
Theoremfuni 5013 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
 |- 
 Fun  _I
 
Theoremnfunv 5014 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
 |- 
 -.  Fun  _V
 
Theoremfunopg 5015 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 )
 
Theoremfunopab 5016* 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 )
 
Theoremfunopabeq 5017* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
 |- 
 Fun  { <. x ,  y >.  |  y  =  A }
 
Theoremfunopab4 5018* A class of ordered pairs of values in the form used by df-mpt 3878 is a function. (Contributed by NM, 17-Feb-2013.)
 |- 
 Fun  { <. x ,  y >.  |  ( ph  /\  y  =  A ) }
 
Theoremfunmpt 5019 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |- 
 Fun  ( x  e.  A  |->  B )
 
Theoremfunmpt2 5020 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
 
Theoremfunco 5021 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 ) )
 
Theoremfunres 5022 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 ) )
 
Theoremfunssres 5023 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 )
 
Theoremfun2ssres 5024 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 ) )
 
Theoremfunun 5025 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 ) )
 
Theoremfuncnvsn 5026 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5029 via cnvsn 4881, 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 >. }
 
Theoremfunsng 5027 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 >. } )
 
Theoremfnsng 5028 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 } )
 
Theoremfunsn 5029 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 >. }
 
Theoremfuninsn 5030 A function based on the singleton of an ordered pair. Unlike funsng 5027, 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 ) )
 
Theoremfunprg 5031 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 >. } )
 
Theoremfuntpg 5032 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 >. } )
 
Theoremfunpr 5033 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 >. } )
 
Theoremfuntp 5034 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 >. } )
 
Theoremfnsn 5035 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 }
 
Theoremfnprg 5036 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 } )
 
Theoremfntpg 5037 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 } )
 
Theoremfntp 5038 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 }
 )
 
Theoremfun0 5039 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
 |- 
 Fun  (/)
 
Theoremfuncnvcnv 5040 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
 |-  ( Fun  A  ->  Fun  `' `' A )
 
Theoremfuncnv2 5041* A simpler equivalence for single-rooted (see funcnv 5042). (Contributed by NM, 9-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y E* x  x A y )
 
Theoremfuncnv 5042* 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 5041 for a simpler version. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
 
Theoremfuncnv3 5043* A condition showing a class is single-rooted. (See funcnv 5042). (Contributed by NM, 26-May-2006.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
 
Theoremfuncnveq 5044* Another way of expressing that a class is single-rooted. Counterpart to dffun2 4993. (Contributed by Jim Kingdon, 24-Dec-2018.)
 |-  ( Fun  `' A  <->  A. x A. y A. z ( ( x A y  /\  z A y )  ->  x  =  z )
 )
 
Theoremfun2cnv 5045* 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 )
 
Theoremsvrelfun 5046 A single-valued relation is a function. (See fun2cnv 5045 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
 |-  ( Fun  A  <->  ( Rel  A  /\  Fun  `' `' A ) )
 
Theoremfncnv 5047* Single-rootedness (see funcnv 5042) 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 )
 
Theoremfun11 5048* 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 ) ) )
 
Theoremfununi 5049* 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 )
 
Theoremfuncnvuni 5050* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5042 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 )
 
Theoremfun11uni 5051* 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 ) )
 
Theoremfunin 5052 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 ) )
 
Theoremfunres11 5053 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
 |-  ( Fun  `' F  ->  Fun  `' ( F  |`  A ) )
 
Theoremfuncnvres 5054 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
 |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A ) ) )
 
Theoremcnvresid 5055 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
 |-  `' (  _I  |`  A )  =  (  _I  |`  A )
 
Theoremfuncnvres2 5056 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 ) ) )
 
Theoremfunimacnv 5057 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
 |-  ( Fun  F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F )
 )
 
Theoremfunimass1 5058 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 ) ) )
 
Theoremfunimass2 5059 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 )
 
Theoremimadiflem 5060 One direction of imadif 5061. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
 |-  ( ( F " A )  \  ( F
 " B ) ) 
 C_  ( F "
 ( A  \  B ) )
 
Theoremimadif 5061 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 ) ) )
 
Theoremimainlem 5062 One direction of imain 5063. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
 |-  ( F " ( A  i^i  B ) ) 
 C_  ( ( F
 " A )  i^i  ( F " B ) )
 
Theoremimain 5063 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
 |-  ( Fun  `' F  ->  ( F " ( A  i^i  B ) )  =  ( ( F
 " A )  i^i  ( F " B ) ) )
 
Theoremfunimaexglem 5064 Lemma for funimaexg 5065. It constitutes the interesting part of funimaexg 5065, in which  B 
C_  dom  A. (Contributed by Jim Kingdon, 27-Dec-2018.)
 |-  ( ( Fun  A  /\  B  e.  C  /\  B  C_  dom  A )  ->  ( A " B )  e.  _V )
 
Theoremfunimaexg 5065 Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
 |-  ( ( Fun  A  /\  B  e.  C ) 
 ->  ( A " B )  e.  _V )
 
Theoremfunimaex 5066 The image of a set under any function is also a set. Equivalent of Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)
 |-  B  e.  _V   =>    |-  ( Fun  A  ->  ( A " B )  e.  _V )
 
Theoremisarep1 5067* Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by  ph ( x ,  y ) i.e. the class  ( {
<. x ,  y >.  |  ph } " A
). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
 |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
 
Theoremisarep2 5068* Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature " [ i, 
[ i, i  ] => o  ] => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5066. (Contributed by NM, 26-Oct-2006.)
 |-  A  e.  _V   &    |-  A. x  e.  A  A. y A. z ( ( ph  /\ 
 [ z  /  y ] ph )  ->  y  =  z )   =>    |- 
 E. w  w  =  ( { <. x ,  y >.  |  ph } " A )
 
Theoremfneq1 5069 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2 5070 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq1d 5071 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2d 5072 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq12d 5073 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  B ) )
 
Theoremfneq12 5074 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ( F  =  G  /\  A  =  B )  ->  ( F  Fn  A 
 <->  G  Fn  B ) )
 
Theoremfneq1i 5075 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F  Fn  A 
 <->  G  Fn  A )
 
Theoremfneq2i 5076 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
 |-  A  =  B   =>    |-  ( F  Fn  A 
 <->  F  Fn  B )
 
Theoremnffn 5077 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/ x  F  Fn  A
 
Theoremfnfun 5078 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Fun  F )
 
Theoremfnrel 5079 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Rel  F )
 
Theoremfndm 5080 The domain of a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F  Fn  A  ->  dom  F  =  A )
 
Theoremfunfni 5081 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
 |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  ph )   =>    |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ph )
 
Theoremfndmu 5082 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( ( F  Fn  A  /\  F  Fn  B )  ->  A  =  B )
 
Theoremfnbr 5083 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
 |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )
 
Theoremfnop 5084 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
 |-  ( ( F  Fn  A  /\  <. B ,  C >.  e.  F )  ->  B  e.  A )
 
Theoremfneu 5085* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
 
Theoremfneu2 5086* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y <. B ,  y >.  e.  F )
 
Theoremfnun 5087 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
 |-  ( ( ( F  Fn  A  /\  G  Fn  B )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B ) )
 
Theoremfnunsn 5088 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  Y  e.  _V )   &    |-  ( ph  ->  F  Fn  D )   &    |-  G  =  ( F  u.  { <. X ,  Y >. } )   &    |-  E  =  ( D  u.  { X } )   &    |-  ( ph  ->  -.  X  e.  D )   =>    |-  ( ph  ->  G  Fn  E )
 
Theoremfnco 5089 Composition of two functions. (Contributed by NM, 22-May-2006.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G )  Fn  B )
 
Theoremfnresdm 5090 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
 |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
 
Theoremfnresdisj 5091 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
 |-  ( F  Fn  A  ->  ( ( A  i^i  B )  =  (/)  <->  ( F  |`  B )  =  (/) ) )
 
Theorem2elresin 5092 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
 |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G )  <->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) )
 
Theoremfnssresb 5093 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
 |-  ( F  Fn  A  ->  ( ( F  |`  B )  Fn  B  <->  B  C_  A ) )
 
Theoremfnssres 5094 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( F  |`  B )  Fn  B )
 
Theoremfnresin1 5095 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
 |-  ( F  Fn  A  ->  ( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )
 
Theoremfnresin2 5096 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
 |-  ( F  Fn  A  ->  ( F  |`  ( B  i^i  A ) )  Fn  ( B  i^i  A ) )
 
Theoremfnres 5097* An equivalence for functionality of a restriction. Compare dffun8 5010. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ( F  |`  A )  Fn  A  <->  A. x  e.  A  E! y  x F y )
 
Theoremfnresi 5098 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
 |-  (  _I  |`  A )  Fn  A
 
Theoremfnima 5099 The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
 
Theoremfn0 5100 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F  Fn  (/)  <->  F  =  (/) )
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