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Theorem fnsingle 25764
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 3474 . . . . 5  |-  ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (  _I  (x)  _V ) ) )  C_  ( _V  X.  _V )
2 df-rel 4885 . . . . 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 201 . . . 4  |-  Rel  (
( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
) )
4 df-singleton 25706 . . . . 5  |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
) )
54releqi 4960 . . . 4  |-  ( Rel Singleton  <->  Rel  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) (  _I  (x)  _V ) ) ) )
63, 5mpbir 201 . . 3  |-  Rel Singleton
7 vex 2959 . . . . . . 7  |-  x  e. 
_V
8 vex 2959 . . . . . . 7  |-  y  e. 
_V
97, 8brsingle 25762 . . . . . 6  |-  ( xSingleton
y  <->  y  =  {
x } )
10 vex 2959 . . . . . . 7  |-  z  e. 
_V
117, 10brsingle 25762 . . . . . 6  |-  ( xSingleton
z  <->  z  =  {
x } )
12 eqtr3 2455 . . . . . 6  |-  ( ( y  =  { x }  /\  z  =  {
x } )  -> 
y  =  z )
139, 11, 12syl2anb 466 . . . . 5  |-  ( ( xSingleton y  /\  xSingleton z )  ->  y  =  z )
1413ax-gen 1555 . . . 4  |-  A. z
( ( xSingleton y  /\  xSingleton z )  -> 
y  =  z )
1514gen2 1556 . . 3  |-  A. x A. y A. z ( ( xSingleton y  /\  xSingleton z )  ->  y  =  z )
16 dffun2 5464 . . 3  |-  ( Fun Singleton  <->  ( Rel Singleton 
/\  A. x A. y A. z ( ( xSingleton
y  /\  xSingleton z )  ->  y  =  z ) ) )
176, 15, 16mpbir2an 887 . 2  |-  Fun Singleton
18 eqv 3643 . . 3  |-  ( dom Singleton  =  _V  <->  A. x  x  e. 
dom Singleton )
19 eqid 2436 . . . . . 6  |-  { x }  =  { x }
20 snex 4405 . . . . . . 7  |-  { x }  e.  _V
217, 20brsingle 25762 . . . . . 6  |-  ( xSingleton { x }  <->  { x }  =  { x } )
2219, 21mpbir 201 . . . . 5  |-  xSingleton { x }
23 breq2 4216 . . . . . 6  |-  ( y  =  { x }  ->  ( xSingleton y  <->  xSingleton { x } ) )
2420, 23spcev 3043 . . . . 5  |-  ( xSingleton { x }  ->  E. y  xSingleton y )
2522, 24ax-mp 8 . . . 4  |-  E. y  xSingleton y
267eldm 5067 . . . 4  |-  ( x  e.  dom Singleton  <->  E. y  xSingleton y
)
2725, 26mpbir 201 . . 3  |-  x  e. 
dom Singleton
2818, 27mpgbir 1559 . 2  |-  dom Singleton  =  _V
29 df-fn 5457 . 2  |-  (Singleton  Fn  _V 
<->  ( Fun Singleton  /\  dom Singleton  =  _V ) )
3017, 28, 29mpbir2an 887 1  |- Singleton  Fn  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    C_ wss 3320   {csn 3814   class class class wbr 4212    _E cep 4492    _I cid 4493    X. cxp 4876   dom cdm 4878   ran crn 4879   Rel wrel 4883   Fun wfun 5448    Fn wfn 5449  (++)csymdif 25662    (x) ctxp 25674  Singletoncsingle 25682
This theorem is referenced by:  fvsingle  25765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-eprel 4494  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-symdif 25663  df-txp 25698  df-singleton 25706
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