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Theorem fvsingle 25757
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 5720 . . . 4  |-  ( x  =  A  ->  (Singleton `  x )  =  (Singleton `  A ) )
2 sneq 3817 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
31, 2eqeq12d 2449 . . 3  |-  ( x  =  A  ->  (
(Singleton `  x )  =  { x }  <->  (Singleton `  A
)  =  { A } ) )
4 eqid 2435 . . . . 5  |-  { x }  =  { x }
5 vex 2951 . . . . . 6  |-  x  e. 
_V
6 snex 4397 . . . . . 6  |-  { x }  e.  _V
75, 6brsingle 25754 . . . . 5  |-  ( xSingleton { x }  <->  { x }  =  { x } )
84, 7mpbir 201 . . . 4  |-  xSingleton { x }
9 fnsingle 25756 . . . . 5  |- Singleton  Fn  _V
10 fnbrfvb 5759 . . . . 5  |-  ( (Singleton  Fn  _V  /\  x  e. 
_V )  ->  (
(Singleton `  x )  =  { x }  <->  xSingleton { x } ) )
119, 5, 10mp2an 654 . . . 4  |-  ( (Singleton `  x )  =  {
x }  <->  xSingleton { x } )
128, 11mpbir 201 . . 3  |-  (Singleton `  x
)  =  { x }
133, 12vtoclg 3003 . 2  |-  ( A  e.  _V  ->  (Singleton `  A )  =  { A } )
14 fvprc 5714 . . 3  |-  ( -.  A  e.  _V  ->  (Singleton `  A )  =  (/) )
15 snprc 3863 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1615biimpi 187 . . 3  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1714, 16eqtr4d 2470 . 2  |-  ( -.  A  e.  _V  ->  (Singleton `  A )  =  { A } )
1813, 17pm2.61i 158 1  |-  (Singleton `  A
)  =  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   {csn 3806   class class class wbr 4204    Fn wfn 5441   ` cfv 5446  Singletoncsingle 25674
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-eprel 4486  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-1st 6341  df-2nd 6342  df-symdif 25655  df-txp 25690  df-singleton 25698
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