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Theorem fvsingle 25485
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 5670 . . 3  |-  ( x  =  A  ->  (Singleton `  x )  =  (Singleton `  A ) )
2 sneq 3770 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
31, 2eqeq12d 2403 . 2  |-  ( x  =  A  ->  (
(Singleton `  x )  =  { x }  <->  (Singleton `  A
)  =  { A } ) )
4 eqid 2389 . . . 4  |-  { x }  =  { x }
5 vex 2904 . . . . 5  |-  x  e. 
_V
6 snex 4348 . . . . 5  |-  { x }  e.  _V
75, 6brsingle 25482 . . . 4  |-  ( xSingleton { x }  <->  { x }  =  { x } )
84, 7mpbir 201 . . 3  |-  xSingleton { x }
9 fnsingle 25484 . . . 4  |- Singleton  Fn  _V
10 fnbrfvb 5708 . . . 4  |-  ( (Singleton  Fn  _V  /\  x  e. 
_V )  ->  (
(Singleton `  x )  =  { x }  <->  xSingleton { x } ) )
119, 5, 10mp2an 654 . . 3  |-  ( (Singleton `  x )  =  {
x }  <->  xSingleton { x } )
128, 11mpbir 201 . 2  |-  (Singleton `  x
)  =  { x }
133, 12vtoclg 2956 1  |-  ( A  e.  V  ->  (Singleton `  A )  =  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2901   {csn 3759   class class class wbr 4155    Fn wfn 5391   ` cfv 5396  Singletoncsingle 25407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-eprel 4437  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-1st 6290  df-2nd 6291  df-symdif 25388  df-txp 25421  df-singleton 25429
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