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Theorem fvsingle 24459
Description: The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvsingle  |-  ( A  e.  V  ->  (Singleton `  A )  =  { A } )

Proof of Theorem fvsingle
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . 3  |-  ( x  =  A  ->  (Singleton `  x )  =  (Singleton `  A ) )
2 sneq 3651 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
31, 2eqeq12d 2297 . 2  |-  ( x  =  A  ->  (
(Singleton `  x )  =  { x }  <->  (Singleton `  A
)  =  { A } ) )
4 eqid 2283 . . . 4  |-  { x }  =  { x }
5 vex 2791 . . . . 5  |-  x  e. 
_V
6 snex 4216 . . . . 5  |-  { x }  e.  _V
75, 6brsingle 24456 . . . 4  |-  ( xSingleton { x }  <->  { x }  =  { x } )
84, 7mpbir 200 . . 3  |-  xSingleton { x }
9 fnsingle 24458 . . . 4  |- Singleton  Fn  _V
10 fnbrfvb 5563 . . . 4  |-  ( (Singleton  Fn  _V  /\  x  e. 
_V )  ->  (
(Singleton `  x )  =  { x }  <->  xSingleton { x } ) )
119, 5, 10mp2an 653 . . 3  |-  ( (Singleton `  x )  =  {
x }  <->  xSingleton { x } )
128, 11mpbir 200 . 2  |-  (Singleton `  x
)  =  { x }
133, 12vtoclg 2843 1  |-  ( A  e.  V  ->  (Singleton `  A )  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   class class class wbr 4023    Fn wfn 5250   ` cfv 5255  Singletoncsingle 24381
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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