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Theorem fvsingle 30735
Description: The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)
Assertion
Ref Expression
fvsingle  |-  (Singleton `  A
)  =  { A }

Proof of Theorem fvsingle
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5887 . . . 4  |-  ( x  =  A  ->  (Singleton `  x )  =  (Singleton `  A ) )
2 sneq 3989 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
31, 2eqeq12d 2476 . . 3  |-  ( x  =  A  ->  (
(Singleton `  x )  =  { x }  <->  (Singleton `  A
)  =  { A } ) )
4 eqid 2461 . . . . 5  |-  { x }  =  { x }
5 vex 3059 . . . . . 6  |-  x  e. 
_V
6 snex 4654 . . . . . 6  |-  { x }  e.  _V
75, 6brsingle 30732 . . . . 5  |-  ( xSingleton { x }  <->  { x }  =  { x } )
84, 7mpbir 214 . . . 4  |-  xSingleton { x }
9 fnsingle 30734 . . . . 5  |- Singleton  Fn  _V
10 fnbrfvb 5927 . . . . 5  |-  ( (Singleton  Fn  _V  /\  x  e. 
_V )  ->  (
(Singleton `  x )  =  { x }  <->  xSingleton { x } ) )
119, 5, 10mp2an 683 . . . 4  |-  ( (Singleton `  x )  =  {
x }  <->  xSingleton { x } )
128, 11mpbir 214 . . 3  |-  (Singleton `  x
)  =  { x }
133, 12vtoclg 3118 . 2  |-  ( A  e.  _V  ->  (Singleton `  A )  =  { A } )
14 fvprc 5881 . . 3  |-  ( -.  A  e.  _V  ->  (Singleton `  A )  =  (/) )
15 snprc 4047 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1615biimpi 199 . . 3  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1714, 16eqtr4d 2498 . 2  |-  ( -.  A  e.  _V  ->  (Singleton `  A )  =  { A } )
1813, 17pm2.61i 169 1  |-  (Singleton `  A
)  =  { A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    = wceq 1454    e. wcel 1897   _Vcvv 3056   (/)c0 3742   {csn 3979   class class class wbr 4415    Fn wfn 5595   ` cfv 5600  Singletoncsingle 30652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-symdif 3674  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-eprel 4763  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fo 5606  df-fv 5608  df-1st 6819  df-2nd 6820  df-txp 30668  df-singleton 30676
This theorem is referenced by: (None)
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