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Theorem fvsingle 29801
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 5848 . . . 4  |-  ( x  =  A  ->  (Singleton `  x )  =  (Singleton `  A ) )
2 sneq 4026 . . . 4  |-  ( x  =  A  ->  { x }  =  { A } )
31, 2eqeq12d 2476 . . 3  |-  ( x  =  A  ->  (
(Singleton `  x )  =  { x }  <->  (Singleton `  A
)  =  { A } ) )
4 eqid 2454 . . . . 5  |-  { x }  =  { x }
5 vex 3109 . . . . . 6  |-  x  e. 
_V
6 snex 4678 . . . . . 6  |-  { x }  e.  _V
75, 6brsingle 29798 . . . . 5  |-  ( xSingleton { x }  <->  { x }  =  { x } )
84, 7mpbir 209 . . . 4  |-  xSingleton { x }
9 fnsingle 29800 . . . . 5  |- Singleton  Fn  _V
10 fnbrfvb 5888 . . . . 5  |-  ( (Singleton  Fn  _V  /\  x  e. 
_V )  ->  (
(Singleton `  x )  =  { x }  <->  xSingleton { x } ) )
119, 5, 10mp2an 670 . . . 4  |-  ( (Singleton `  x )  =  {
x }  <->  xSingleton { x } )
128, 11mpbir 209 . . 3  |-  (Singleton `  x
)  =  { x }
133, 12vtoclg 3164 . 2  |-  ( A  e.  _V  ->  (Singleton `  A )  =  { A } )
14 fvprc 5842 . . 3  |-  ( -.  A  e.  _V  ->  (Singleton `  A )  =  (/) )
15 snprc 4079 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1615biimpi 194 . . 3  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1714, 16eqtr4d 2498 . 2  |-  ( -.  A  e.  _V  ->  (Singleton `  A )  =  { A } )
1813, 17pm2.61i 164 1  |-  (Singleton `  A
)  =  { A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   {csn 4016   class class class wbr 4439    Fn wfn 5565   ` cfv 5570  Singletoncsingle 29718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-symdif 3715  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-eprel 4780  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773  df-2nd 6774  df-txp 29734  df-singleton 29742
This theorem is referenced by: (None)
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