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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnvso 5201 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)
 |-  ( R  Or  A  <->  `' R  Or  A )
 
Theoremcoexg 5202 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  o.  B )  e.  _V )
 
Theoremcoex 5203 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  o.  B )  e.  _V
 
Theoremdffun2 5204* 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 )
 ) )
 
Theoremdffun3 5205* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
 |-  ( Fun  A  <->  ( Rel  A  /\  A. x E. z A. y ( x A y  ->  y  =  z ) ) )
 
Theoremdffun4 5206* 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 )
 ) )
 
Theoremdffun5 5207* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
 |-  ( Fun  A  <->  ( Rel  A  /\  A. x E. z A. y ( <. x ,  y >.  e.  A  ->  y  =  z ) ) )
 
Theoremdffun6f 5208* 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 5209* 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 5210* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
 |-  ( Fun  F  ->  E* y  A F y )
 
Theoremfunrel 5211 A function is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( Fun  A  ->  Rel 
 A )
 
Theoremfunss 5212 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 5213 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremfuneqi 5214 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  A  =  B   =>    |-  ( Fun  A  <->  Fun 
 B )
 
Theoremfuneqd 5215 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremnffun 5216 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
 |-  F/_ x F   =>    |- 
 F/ x Fun  F
 
Theoremfuneu 5217* 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 5218* 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 5219* 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 5220 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 5220* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5219. (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 5221* 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 5222 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  A  <->  A  Fn  dom  A )
 
Theoremfuni 5223 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
 |- 
 Fun  _I
 
Theoremnfunv 5224 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
 |- 
 -.  Fun  _V
 
Theoremfunopg 5225 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 5226* 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 5227* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
 |- 
 Fun  { <. x ,  y >.  |  y  =  A }
 
Theoremfunopab4 5228* A class of ordered pairs of values in the form used by df-mpt 4053 is a function. (Contributed by NM, 17-Feb-2013.)
 |- 
 Fun  { <. x ,  y >.  |  ( ph  /\  y  =  A ) }
 
Theoremfunmpt 5229 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |- 
 Fun  ( x  e.  A  |->  B )
 
Theoremfunco 5230 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 5231 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 5232 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 5233 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 5234 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 5235 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5238 via cnvsn 5142, 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 5236 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 5237 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 5238 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 >. }
 
Theoremfunprg 5239 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 >. } )
 
Theoremfunpr 5240 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 5241 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 5242 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 5243 Domain of a 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 } )
 
Theoremfntp 5244 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 5245 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
 |- 
 Fun  (/)
 
Theoremfuncnvcnv 5246 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
 |-  ( Fun  A  ->  Fun  `' `' A )
 
Theoremfuncnv2 5247* A simpler equivalence for single-rooted (see funcnv 5248). (Contributed by NM, 9-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y E* x  x A y )
 
Theoremfuncnv 5248* 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 5247 for a simpler version. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
 
Theoremfuncnv3 5249* A condition showing a class is single-rooted. (See funcnv 5248). (Contributed by NM, 26-May-2006.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
 
Theoremfun2cnv 5250* 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 5251 A single-valued relation is a function. (See fun2cnv 5250 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
 |-  ( Fun  A  <->  ( Rel  A  /\  Fun  `' `' A ) )
 
Theoremfncnv 5252* Single-rootedness (see funcnv 5248) 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 5253* 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 5254* 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 5255* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5248 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 5256* 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 5257 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 5258 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
 |-  ( Fun  `' F  ->  Fun  `' ( F  |`  A ) )
 
Theoremfuncnvres 5259 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
 |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A ) ) )
 
Theoremcnvresid 5260 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
 |-  `' (  _I  |`  A )  =  (  _I  |`  A )
 
Theoremfuncnvres2 5261 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 5262 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 5263 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 5264 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 )
 
Theoremimadif 5265 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 ) ) )
 
Theoremimain 5266 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 ) ) )
 
Theoremfunimaexg 5267 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 5268 The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4105. 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 5269* 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 5270* 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 5268. (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 5271 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2 5272 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq1d 5273 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 5274 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 5275 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 ) )
 
Theoremfneq1i 5276 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F  Fn  A 
 <->  G  Fn  A )
 
Theoremfneq2i 5277 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
 |-  A  =  B   =>    |-  ( F  Fn  A 
 <->  F  Fn  B )
 
Theoremnffn 5278 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 5279 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Fun  F )
 
Theoremfnrel 5280 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Rel  F )
 
Theoremfndm 5281 The domain of a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F  Fn  A  ->  dom  F  =  A )
 
Theoremfunfni 5282 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 5283 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( ( F  Fn  A  /\  F  Fn  B )  ->  A  =  B )
 
Theoremfnbr 5284 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 5285 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 5286* 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 5287* 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 5288 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 5289 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 5290 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 5291 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
 |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
 
Theoremfnresdisj 5292 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 5293 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 5294 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 5295 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 5296 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 5297 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 5298* An equivalence for functionality of a restriction. Compare dffun8 5220. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ( F  |`  A )  Fn  A  <->  A. x  e.  A  E! y  x F y )
 
Theoremfnresi 5299 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
 |-  (  _I  |`  A )  Fn  A
 
Theoremfnima 5300 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 )
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