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Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-iota 5301* 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 5312); otherwise, it evaluates to the empty set (see iotanul 5316). 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 6405 (or iotacl 5324 for unbounded iota), as demonstrated in the proof of supub 7300. This can be easier than applying riotasbc 6407 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 5302* 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 5303 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 5304 Deduction version of nfiota 5305. (Contributed by NM, 18-Feb-2013.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/_ x ( iota y ps ) )
 
Theoremnfiota 5305 Bound-variable hypothesis builder for the  iota class. (Contributed by NM, 23-Aug-2011.)
 |- 
 F/ x ph   =>    |-  F/_ x ( iota y ph )
 
Theoremcbviota 5306 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 5307* 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 5308 Variable substitution in description binder. Compare sb8eu 2227. (Contributed by NM, 18-Mar-2013.)
 |- 
 F/ y ph   =>    |-  ( iota x ph )  =  ( iota y [ y  /  x ] ph )
 
Theoremiotaeq 5309 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
 
Theoremiotabi 5310 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
 
Theoremuniabio 5311* 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 5312* 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 5313 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  =  U. { x  |  ph } )
 
Theoremiotaint 5314 Equivalence between two different forms of  iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E! x ph  ->  ( iota x ph )  =  |^| { x  |  ph } )
 
Theoremiota1 5315 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 5316 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 5317 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 5318 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 5319 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  -> 
 [. ( iota x ph )  /  x ]. ph )
 
Theoremiota4an 5320 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 5321* 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 5322* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 iota x ps )  =  ( iota x ch ) )
 
Theoremiotabii 5323 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( iota x ph )  =  ( iota x ps )
 
Theoremiotacl 5324 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 5301). If you have a bounded iota-based definition, riotacl2 6405 may be useful.

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

 |-  ( E! x ph  ->  ( iota x ph )  e.  { x  |  ph } )
 
Theoremiota2df 5325 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 5326* 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 5327* 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 5328 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 5329 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 5330* 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 5331 Extend the definition of a wff to include the function predicate. (Read:  A is a function.)
 wff  Fun  A
 
Syntaxwfn 5332 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 5333 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 5334 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 5335 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 5336 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 5337 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 5338 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 5339 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 12449). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4158 with the maps-to notation (see df-mpt 4160 and df-mpt2 5950). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5340), a function with a given domain and codomain (df-f 5341), a one-to-one function (df-f1 5342), an onto function (df-fo 5343), or a one-to-one onto function (df-f1o 5344). For alternate definitions, see dffun2 5347, dffun3 5348, dffun4 5349, dffun5 5350, dffun6 5352, dffun7 5362, dffun8 5363, and dffun9 5364. (Contributed by NM, 1-Aug-1994.)
 |-  ( Fun  A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  )
 )
 
Definitiondf-fn 5340 Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. For alternate definitions, see dffn2 5473, dffn3 5479, dffn4 5540, and dffn5 5651. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  Fn  B  <->  ( Fun  A  /\  dom  A  =  B ) )
 
Definitiondf-f 5341 Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. For alternate definitions, see dff2 5755, dff3 5756, and dff4 5757. (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A --> B 
 <->  ( F  Fn  A  /\  ran  F  C_  B ) )
 
Definitiondf-f1 5342 Define a one-to-one function. For equivalent definitions see dff12 5519 and dff13 5870. 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 5343 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 5538, dffo3 5758, dffo4 5759, and dffo5 5760. (Contributed by NM, 1-Aug-1994.)
 |-  ( F : A -onto-> B 
 <->  ( F  Fn  A  /\  ran  F  =  B ) )
 
Definitiondf-f1o 5344 Define a one-to-one onto function. For equivalent definitions see dff1o2 5560, dff1o3 5561, dff1o4 5563, and dff1o5 5564. 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 5345* Define the value of a function,  ( F `  A
), also known as function application. For example,  ( cos `  0
)  =  1 (we prove this in cos0 12527 after we define cosine in df-cos 12449). Typically, function  F is defined using maps-to notation (see df-mpt 4160 and df-mpt2 5950), but this is not required. For example,  F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
>. }  ->  ( F `  3 )  =  9 (ex-fv 20942). Note that df-ov 5948 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 5635 and fvprc 5602). 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 5675, dffv3 5604, fv2 5603, and fv3 5624 (the latter two previously required  A to be a set.) Restricted equivalents that require  F to be a function are shown in funfv 5669 and funfv2 5670. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 5649. (Contributed by Scott Fenton, 6-Oct-2017.)
 |-  ( F `  A )  =  ( iota x A F x )
 
Definitiondf-isom 5346* 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 5347* 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 5348* 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 5349* 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 5350* 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 5351* 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 5352* 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 5353* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)
 |-  ( Fun  F  ->  E* y  A F y )
 
Theoremfunrel 5354 A function is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( Fun  A  ->  Rel 
 A )
 
Theoremfunss 5355 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 5356 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremfuneqi 5357 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  A  =  B   =>    |-  ( Fun  A  <->  Fun 
 B )
 
Theoremfuneqd 5358 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )
 
Theoremnffun 5359 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
 |-  F/_ x F   =>    |- 
 F/ x Fun  F
 
Theoremfuneu 5360* 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 5361* 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 5362* 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 5363 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 5363* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5362. (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 5364* 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 5365 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  A  <->  A  Fn  dom  A )
 
Theoremfuni 5366 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)
 |- 
 Fun  _I
 
Theoremnfunv 5367 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
 |- 
 -.  Fun  _V
 
Theoremfunopg 5368 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 5369* 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 5370* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
 |- 
 Fun  { <. x ,  y >.  |  y  =  A }
 
Theoremfunopab4 5371* A class of ordered pairs of values in the form used by df-mpt 4160 is a function. (Contributed by NM, 17-Feb-2013.)
 |- 
 Fun  { <. x ,  y >.  |  ( ph  /\  y  =  A ) }
 
Theoremfunmpt 5372 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |- 
 Fun  ( x  e.  A  |->  B )
 
Theoremfunmpt2 5373 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 5374 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 5375 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 5376 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 5377 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 5378 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 5379 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5382 via cnvsn 5237, 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 5380 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 5381 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 5382 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 5383 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 5384 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 5385 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 5386 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 5387 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 5388 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 5389 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
 |- 
 Fun  (/)
 
Theoremfuncnvcnv 5390 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
 |-  ( Fun  A  ->  Fun  `' `' A )
 
Theoremfuncnv2 5391* A simpler equivalence for single-rooted (see funcnv 5392). (Contributed by NM, 9-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y E* x  x A y )
 
Theoremfuncnv 5392* 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 5391 for a simpler version. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
 
Theoremfuncnv3 5393* A condition showing a class is single-rooted. (See funcnv 5392). (Contributed by NM, 26-May-2006.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
 
Theoremfun2cnv 5394* 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 5395 A single-valued relation is a function. (See fun2cnv 5394 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
 |-  ( Fun  A  <->  ( Rel  A  /\  Fun  `' `' A ) )
 
Theoremfncnv 5396* Single-rootedness (see funcnv 5392) 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 5397* 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 5398* 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 5399* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5392 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 5400* 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 ) )
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