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Theorem fnsingle 24529
Description: The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fnsingle  |- Singleton  Fn  _V

Proof of Theorem fnsingle
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difss 3316 . . . . 5  |-  ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (  _I  (x)  _V ) ) )  C_  ( _V  X.  _V )
2 df-rel 4712 . . . . 5  |-  ( Rel  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) (  _I  (x)  _V ) ) )  <->  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V ) ) )  C_  ( _V  X.  _V )
)
31, 2mpbir 200 . . . 4  |-  Rel  (
( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
) )
4 df-singleton 24474 . . . . 5  |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
) )
54releqi 4788 . . . 4  |-  ( Rel Singleton  <->  Rel  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) (  _I  (x)  _V ) ) ) )
63, 5mpbir 200 . . 3  |-  Rel Singleton
7 vex 2804 . . . . . . 7  |-  x  e. 
_V
8 vex 2804 . . . . . . 7  |-  y  e. 
_V
97, 8brsingle 24527 . . . . . 6  |-  ( xSingleton
y  <->  y  =  {
x } )
10 vex 2804 . . . . . . 7  |-  z  e. 
_V
117, 10brsingle 24527 . . . . . 6  |-  ( xSingleton
z  <->  z  =  {
x } )
12 eqtr3 2315 . . . . . 6  |-  ( ( y  =  { x }  /\  z  =  {
x } )  -> 
y  =  z )
139, 11, 12syl2anb 465 . . . . 5  |-  ( ( xSingleton y  /\  xSingleton z )  ->  y  =  z )
1413ax-gen 1536 . . . 4  |-  A. z
( ( xSingleton y  /\  xSingleton z )  -> 
y  =  z )
1514gen2 1537 . . 3  |-  A. x A. y A. z ( ( xSingleton y  /\  xSingleton z )  ->  y  =  z )
16 dffun2 5281 . . 3  |-  ( Fun Singleton  <->  ( Rel Singleton 
/\  A. x A. y A. z ( ( xSingleton
y  /\  xSingleton z )  ->  y  =  z ) ) )
176, 15, 16mpbir2an 886 . 2  |-  Fun Singleton
18 eqv 3483 . . 3  |-  ( dom Singleton  =  _V  <->  A. x  x  e. 
dom Singleton )
19 eqid 2296 . . . . . 6  |-  { x }  =  { x }
20 snex 4232 . . . . . . 7  |-  { x }  e.  _V
217, 20brsingle 24527 . . . . . 6  |-  ( xSingleton { x }  <->  { x }  =  { x } )
2219, 21mpbir 200 . . . . 5  |-  xSingleton { x }
23 breq2 4043 . . . . . 6  |-  ( y  =  { x }  ->  ( xSingleton y  <->  xSingleton { x } ) )
2420, 23spcev 2888 . . . . 5  |-  ( xSingleton { x }  ->  E. y  xSingleton y )
2522, 24ax-mp 8 . . . 4  |-  E. y  xSingleton y
267eldm 4892 . . . 4  |-  ( x  e.  dom Singleton  <->  E. y  xSingleton y
)
2725, 26mpbir 200 . . 3  |-  x  e. 
dom Singleton
2818, 27mpgbir 1540 . 2  |-  dom Singleton  =  _V
29 df-fn 5274 . 2  |-  (Singleton  Fn  _V 
<->  ( Fun Singleton  /\  dom Singleton  =  _V ) )
3017, 28, 29mpbir2an 886 1  |- Singleton  Fn  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039    _E cep 4319    _I cid 4320    X. cxp 4703   dom cdm 4705   ran crn 4706   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266  (++)csymdif 24432    (x) ctxp 24444  Singletoncsingle 24452
This theorem is referenced by:  fvsingle  24530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-singleton 24474
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