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Theorem fnsingle 24458
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 3303 . . . . 5  |-  ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (  _I  (x)  _V ) ) )  C_  ( _V  X.  _V )
2 df-rel 4696 . . . . 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 24403 . . . . 5  |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
) )
54releqi 4772 . . . 4  |-  ( Rel Singleton  <->  Rel  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) (  _I  (x)  _V ) ) ) )
63, 5mpbir 200 . . 3  |-  Rel Singleton
7 vex 2791 . . . . . . 7  |-  x  e. 
_V
8 vex 2791 . . . . . . 7  |-  y  e. 
_V
97, 8brsingle 24456 . . . . . 6  |-  ( xSingleton
y  <->  y  =  {
x } )
10 vex 2791 . . . . . . 7  |-  z  e. 
_V
117, 10brsingle 24456 . . . . . 6  |-  ( xSingleton
z  <->  z  =  {
x } )
12 eqtr3 2302 . . . . . 6  |-  ( ( y  =  { x }  /\  z  =  {
x } )  -> 
y  =  z )
139, 11, 12syl2anb 465 . . . . 5  |-  ( ( xSingleton y  /\  xSingleton z )  ->  y  =  z )
1413ax-gen 1533 . . . 4  |-  A. z
( ( xSingleton y  /\  xSingleton z )  -> 
y  =  z )
1514gen2 1534 . . 3  |-  A. x A. y A. z ( ( xSingleton y  /\  xSingleton z )  ->  y  =  z )
16 dffun2 5265 . . 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 3470 . . 3  |-  ( dom Singleton  =  _V  <->  A. x  x  e. 
dom Singleton )
19 eqid 2283 . . . . . 6  |-  { x }  =  { x }
20 snex 4216 . . . . . . 7  |-  { x }  e.  _V
217, 20brsingle 24456 . . . . . 6  |-  ( xSingleton { x }  <->  { x }  =  { x } )
2219, 21mpbir 200 . . . . 5  |-  xSingleton { x }
23 breq2 4027 . . . . . 6  |-  ( y  =  { x }  ->  ( xSingleton y  <->  xSingleton { x } ) )
2420, 23spcev 2875 . . . . 5  |-  ( xSingleton { x }  ->  E. y  xSingleton y )
2522, 24ax-mp 8 . . . 4  |-  E. y  xSingleton y
267eldm 4876 . . . 4  |-  ( x  e.  dom Singleton  <->  E. y  xSingleton y
)
2725, 26mpbir 200 . . 3  |-  x  e. 
dom Singleton
2818, 27mpgbir 1537 . 2  |-  dom Singleton  =  _V
29 df-fn 5258 . 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 1527   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023    _E cep 4303    _I cid 4304    X. cxp 4687   dom cdm 4689   ran crn 4690   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250  (++)csymdif 24361    (x) ctxp 24373  Singletoncsingle 24381
This theorem is referenced by:  fvsingle  24459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-singleton 24403
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