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Theorem List for Metamath Proof Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelresfld 5201 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
 |-  ( Rel  R  ->  ( R  |`  U. U. R )  =  R )
 
Theoremrelcoi2 5202 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
 |-  ( Rel  R  ->  ( (  _I  |`  U. U. R )  o.  R )  =  R )
 
Theoremrelcoi1 5203 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
 |-  ( Rel  R  ->  ( R  o.  (  _I  |`  U. U. R ) )  =  R )
 
Theoremunidmrn 5204 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
 |- 
 U. U. `' A  =  ( dom  A  u.  ran  A )
 
Theoremrelcnvfld 5205 if  R is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
 |-  ( Rel  R  ->  U.
 U. R  =  U. U. `' R )
 
Theoremdfdm2 5206 Alternate definition of domain df-dm 4701 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
 |- 
 dom  A  =  U. U. ( `' A  o.  A )
 
Theoremunixp 5207 The double class union of a non-empty cross product is the union of it members. (Contributed by NM, 17-Sep-2006.)
 |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B ) )
 
Theoremunixp0 5208 A cross product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
 |-  ( ( A  X.  B )  =  (/)  <->  U. ( A  X.  B )  =  (/) )
 
Theoremunixpid 5209 Field of a square cross product. (Contributed by FL, 10-Oct-2009.)
 |- 
 U. U. ( A  X.  A )  =  A
 
Theoremcnvexg 5210 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
 |-  ( A  e.  V  ->  `' A  e.  _V )
 
Theoremcnvex 5211 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
 |-  A  e.  _V   =>    |-  `' A  e.  _V
 
Theoremrelcnvexb 5212 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
 |-  ( Rel  R  ->  ( R  e.  _V  <->  `' R  e.  _V ) )
 
Theoremressn 5213 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B }
 ) )
 
Theoremcnviin 5214* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
 |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
 x  e.  A  `' B )
 
Theoremcnvpo 5215 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
 |-  ( R  Po  A  <->  `' R  Po  A )
 
Theoremcnvso 5216 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 5217 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 5218 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
 
2.4.8  Definite description binder (inverted iota)
 
Syntaxcio 5219 Extend class notation with Russell's definition description binder (inverted iota).
 class  ( iota x ph )
 
Theoremiotajust 5220* Soundness justification theorem for df-iota 5221. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |- 
 U. { y  |  { x  |  ph }  =  { y } }  =  U. { z  |  { x  |  ph }  =  { z } }
 
Definitiondf-iota 5221* Define Russell's definition description binder, which can be read as "the unique  x such that  ph," where  ph ordinarily contains  x as a free variable. Our definition is meaningful only when there is exactly one  x such that  ph is true (see iotaval 5232); otherwise, it evaluates to the empty set (see iotanul 5236). Russell used the inverted iota symbol 
iota to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use riotacl2 6320 (or iotacl 5244 for unbounded iota), as demonstrated in the proof of supub 7212. This can be easier than applying riotasbc 6322 or a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

 |-  ( iota x ph )  =  U. { y  |  { x  |  ph }  =  { y } }
 
Theoremdfiota2 5222* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( iota x ph )  =  U. { y  |  A. x ( ph  <->  x  =  y ) }
 
Theoremnfiota1 5223 Bound-variable hypothesis builder for the  iota class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x ( iota x ph )
 
Theoremnfiotad 5224 Deduction version of nfiota 5225. (Contributed by NM, 18-Feb-2013.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x ( iota y ps ) )
 
Theoremnfiota 5225 Bound-variable hypothesis builder for the  iota class. (Contributed by NM, 23-Aug-2011.)
 |- 
 F/ x ph   =>    |-  F/_ x ( iota y ph )
 
Theoremcbviota 5226 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( iota x ph )  =  ( iota y ps )
 
Theoremcbviotav 5227* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( iota x ph )  =  ( iota
 y ps )
 
Theoremsb8iota 5228 Variable substitution in description binder. Compare sb8eu 2163. (Contributed by NM, 18-Mar-2013.)
 |- 
 F/ y ph   =>    |-  ( iota x ph )  =  ( iota y [ y  /  x ] ph )
 
Theoremiotaeq 5229 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
 
Theoremiotabi 5230 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
 
Theoremuniabio 5231* Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x (
 ph 
 <->  x  =  y ) 
 ->  U. { x  |  ph
 }  =  y )
 
Theoremiotaval 5232* 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 )
 
Theoremiotauni 5233 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  =  U. { x  |  ph } )
 
Theoremiotaint 5234 Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x ph  ->  ( iota x ph )  =  |^| { x  |  ph } )
 
Theoremiota1 5235 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  ( E! x ph  ->  ( ph  <->  ( iota x ph )  =  x ) )
 
Theoremiotanul 5236 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 )  =  (/) )
 
Theoremiotassuni 5237 The  iota class is a subset of the union of all elements satisfying  ph. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( iota x ph )  C_  U. { x  |  ph }
 
Theoremiotaex 5238 Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the  iota class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x ph )  e.  _V
 
Theoremiota4 5239 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  -> 
 [. ( iota x ph )  /  x ]. ph )
 
Theoremiota4an 5240 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x (
 ph  /\  ps )  -> 
 [. ( iota x ( ph  /\  ps )
 )  /  x ]. ph )
 
Theoremiota5 5241* 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 )
 
Theoremiotabidv 5242* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota x ps )  =  ( iota x ch ) )
 
Theoremiotabii 5243 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( iota x ph )  =  ( iota x ps )
 
Theoremiotacl 5244 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5221). If you have a bounded iota-based definition, riotacl2 6320 may be useful.

(Contributed by Andrew Salmon, 1-Aug-2011.)

 |-  ( E! x ph  ->  ( iota x ph )  e.  { x  |  ph } )
 
Theoremiota2df 5245 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 ) )
 
Theoremiota2d 5246* 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 ) )
 
Theoremiota2 5247* 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 )
 )
 
Theoremsniota 5248 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 ) } )
 
Theoremdfiota4 5249 The  iota operation using the  if operator. (Contributed by Scott Fenton, 6-Oct-2017.)
 |-  ( iota x ph )  =  if ( E! x ph ,  U. { x  |  ph } ,  (/) )
 
Theoremcsbiotag 5250* 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 ) )
 
2.4.9  Functions
 
Syntaxwfun 5251 Extend the definition of a wff to include the function predicate. (Read:  A is a function.)
 wff  Fun  A
 
Syntaxwfn 5252 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
 
Syntaxwf 5253 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
 
Syntaxwf1 5254 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
 
Syntaxwfo 5255 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
 
Syntaxwf1o 5256 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
 
Syntaxcfv 5257 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 )
 
Syntaxwiso 5258 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 )
 
Definitiondf-fun 5259 Define predicate that determines if some class  A is a function. Definition 10.1 of [Quine] p. 65. For example, the expression  Fun  cos is true once we define cosine (df-cos 12354). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4079 with the maps-to notation (see df-mpt 4081 and df-mpt2 5865). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5260), a function with a given domain and codomain (df-f 5261), a one-to-one function (df-f1 5262), an onto function (df-fo 5263), or a one-to-one onto function (df-f1o 5264). For alternate definitions, see dffun2 5267, dffun3 5268, dffun4 5269, dffun5 5270, dffun6 5272, dffun7 5282, dffun8 5283, and dffun9 5284. (Contributed by NM, 1-Aug-1994.)
 |-  ( Fun  A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  )
 )
 
Definitiondf-fn 5260 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 5392, dffn3 5398, dffn4 5459, and dffn5 5570. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  Fn  B  <->  ( Fun  A  /\  dom  A  =  B ) )
 
Definitiondf-f 5261 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 5674, dff3 5675, and dff4 5676. (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  ran  F  C_  B ) )
 
Definitiondf-f1 5262 Define a one-to-one function. For equivalent definitions see dff12 5438 and dff13 5785. 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 ) )
 
Definitiondf-fo 5263 Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). For alternate definitions, see dffo2 5457, dffo3 5677, dffo4 5678, and dffo5 5679. (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A -onto-> B 
 <->  ( F  Fn  A  /\  ran  F  =  B ) )
 
Definitiondf-f1o 5264 Define a one-to-one onto function. For equivalent definitions see dff1o2 5479, dff1o3 5480, dff1o4 5482, and dff1o5 5483. 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 ) )
 
Definitiondf-fv 5265* Define the value of a function,  ( F `  A
), also known as function application. For example,  ( cos `  0
)  =  1 (we prove this in cos0 12432 after we define cosine in df-cos 12354). Typically, function  F is defined using maps-to notation (see df-mpt 4081 and df-mpt2 5865), but this is not required. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ( F `  3 )  =  9 (ex-fv 20832). Note that df-ov 5863 will 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 (as shown by ndmfv 5554 and fvprc 5521). 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. Alternate definitions are dffv2 5594, dffv3 5523, fv2 5522, and fv3 5543 (the latter two previously required  A to be a set.) Restricted equivalents that require  F to be a function are shown in funfv 5588 and funfv2 5589. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5568. (Contributed by Scott Fenton, 6-Oct-2017.)
 |-  ( F `  A )  =  ( iota x A F x )
 
Definitiondf-isom 5266* 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 5267* 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 5268* 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 5269* 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 5270* 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 5271* 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 5272* 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 5273* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
 |-  ( Fun  F  ->  E* y  A F y )
 
Theoremfunrel 5274 A function is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( Fun  A  ->  Rel 
 A )
 
Theoremfunss 5275 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 5276 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremfuneqi 5277 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  A  =  B   =>    |-  ( Fun  A  <->  Fun 
 B )
 
Theoremfuneqd 5278 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremnffun 5279 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
 |-  F/_ x F   =>    |- 
 F/ x Fun  F
 
Theoremfuneu 5280* 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 5281* 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 5282* 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 5283 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 5283* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5282. (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 5284* 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 5285 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  A  <->  A  Fn  dom  A )
 
Theoremfuni 5286 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
 |- 
 Fun  _I
 
Theoremnfunv 5287 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
 |- 
 -.  Fun  _V
 
Theoremfunopg 5288 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 5289* 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 5290* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
 |- 
 Fun  { <. x ,  y >.  |  y  =  A }
 
Theoremfunopab4 5291* A class of ordered pairs of values in the form used by df-mpt 4081 is a function. (Contributed by NM, 17-Feb-2013.)
 |- 
 Fun  { <. x ,  y >.  |  ( ph  /\  y  =  A ) }
 
Theoremfunmpt 5292 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |- 
 Fun  ( x  e.  A  |->  B )
 
Theoremfunmpt2 5293 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 5294 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 5295 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 5296 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 5297 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 5298 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 5299 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5302 via cnvsn 5157, 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 5300 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 >. } )
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