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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-fv 5101* 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 3961), 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 )
 
Definitiondf-isom 5102* 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 ) ) ) )
 
Theoremdffun2 5103* 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 )
 ) )
 
Theoremdffun4 5104* 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 )
 ) )
 
Theoremdffun5r 5105* A way of proving a relation is a function, analogous to mo2r 2029. (Contributed by Jim Kingdon, 27-May-2020.)
 |-  ( ( Rel  A  /\  A. x E. z A. y ( <. x ,  y >.  e.  A  ->  y  =  z ) ) 
 ->  Fun  A )
 
Theoremdffun6f 5106* 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 ) )
 
Theoremdffun6 5107* 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 ) )
 
Theoremfunmo 5108* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
 |-  ( Fun  F  ->  E* y  A F y )
 
Theoremdffun4f 5109* Definition of function like dffun4 5104 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 )
 ) )
 
Theoremfunrel 5110 A function is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( Fun  A  ->  Rel 
 A )
 
Theorem0nelfun 5111 A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
 |-  ( Fun  R  ->  (/)  e/  R )
 
Theoremfunss 5112 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 5113 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremfuneqi 5114 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  A  =  B   =>    |-  ( Fun  A  <->  Fun 
 B )
 
Theoremfuneqd 5115 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremnffun 5116 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
 |-  F/_ x F   =>    |- 
 F/ x Fun  F
 
Theoremsbcfung 5117 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 5118* 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 5119* 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 5120* 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 5121 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 5121* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5120. (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 5122* 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 5123 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  A  <->  A  Fn  dom  A )
 
Theoremfunfnd 5124 A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
 |-  ( ph  ->  Fun  A )   =>    |-  ( ph  ->  A  Fn  dom  A )
 
Theoremfuni 5125 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
 |- 
 Fun  _I
 
Theoremnfunv 5126 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
 |- 
 -.  Fun  _V
 
Theoremfunopg 5127 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 5128* 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 5129* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
 |- 
 Fun  { <. x ,  y >.  |  y  =  A }
 
Theoremfunopab4 5130* A class of ordered pairs of values in the form used by df-mpt 3961 is a function. (Contributed by NM, 17-Feb-2013.)
 |- 
 Fun  { <. x ,  y >.  |  ( ph  /\  y  =  A ) }
 
Theoremfunmpt 5131 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |- 
 Fun  ( x  e.  A  |->  B )
 
Theoremfunmpt2 5132 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 5133 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 5134 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 5135 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 5136 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 5137 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 5138 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5141 via cnvsn 4991, 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 5139 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 5140 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 5141 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 5142 A function based on the singleton of an ordered pair. Unlike funsng 5139, 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 5143 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 5144 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 5145 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 5146 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 5147 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 5148 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 5149 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 5150 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 5151 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
 |- 
 Fun  (/)
 
Theoremfuncnvcnv 5152 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
 |-  ( Fun  A  ->  Fun  `' `' A )
 
Theoremfuncnv2 5153* A simpler equivalence for single-rooted (see funcnv 5154). (Contributed by NM, 9-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y E* x  x A y )
 
Theoremfuncnv 5154* 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 5153 for a simpler version. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
 
Theoremfuncnv3 5155* A condition showing a class is single-rooted. (See funcnv 5154). (Contributed by NM, 26-May-2006.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
 
Theoremfuncnveq 5156* Another way of expressing that a class is single-rooted. Counterpart to dffun2 5103. (Contributed by Jim Kingdon, 24-Dec-2018.)
 |-  ( Fun  `' A  <->  A. x A. y A. z ( ( x A y  /\  z A y )  ->  x  =  z )
 )
 
Theoremfun2cnv 5157* 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 5158 A single-valued relation is a function. (See fun2cnv 5157 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
 |-  ( Fun  A  <->  ( Rel  A  /\  Fun  `' `' A ) )
 
Theoremfncnv 5159* Single-rootedness (see funcnv 5154) 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 5160* 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 5161* 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 5162* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5154 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 5163* 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 5164 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 5165 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
 |-  ( Fun  `' F  ->  Fun  `' ( F  |`  A ) )
 
Theoremfuncnvres 5166 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
 |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A ) ) )
 
Theoremcnvresid 5167 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
 |-  `' (  _I  |`  A )  =  (  _I  |`  A )
 
Theoremfuncnvres2 5168 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 5169 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 5170 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 5171 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 5172 One direction of imadif 5173. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
 |-  ( ( F " A )  \  ( F
 " B ) ) 
 C_  ( F "
 ( A  \  B ) )
 
Theoremimadif 5173 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 5174 One direction of imain 5175. 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 5175 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 5176 Lemma for funimaexg 5177. It constitutes the interesting part of funimaexg 5177, 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 5177 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 5178 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 5179* 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 5180* 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 5178. (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 5181 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2 5182 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq1d 5183 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 5184 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 5185 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 5186 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 5187 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F  Fn  A 
 <->  G  Fn  A )
 
Theoremfneq2i 5188 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
 |-  A  =  B   =>    |-  ( F  Fn  A 
 <->  F  Fn  B )
 
Theoremnffn 5189 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 5190 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Fun  F )
 
Theoremfnrel 5191 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Rel  F )
 
Theoremfndm 5192 The domain of a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F  Fn  A  ->  dom  F  =  A )
 
Theoremfunfni 5193 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 5194 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( ( F  Fn  A  /\  F  Fn  B )  ->  A  =  B )
 
Theoremfnbr 5195 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 5196 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 5197* 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 5198* 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 5199 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 5200 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 )
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