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Theorem brsingle 24456
Description: The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brsingle.1  |-  A  e. 
_V
brsingle.2  |-  B  e. 
_V
Assertion
Ref Expression
brsingle  |-  ( ASingleton B 
<->  B  =  { A } )

Proof of Theorem brsingle
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 brsingle.1 . 2  |-  A  e. 
_V
2 brsingle.2 . 2  |-  B  e. 
_V
3 df-singleton 24403 . 2  |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) (  _I  (x)  _V )
) )
4 brxp 4720 . . 3  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
51, 2, 4mpbir2an 886 . 2  |-  A ( _V  X.  _V ) B
6 elsn 3655 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
71ideq 4836 . . 3  |-  ( x  _I  A  <->  x  =  A )
86, 7bitr4i 243 . 2  |-  ( x  e.  { A }  <->  x  _I  A )
91, 2, 3, 5, 8brtxpsd3 24436 1  |-  ( ASingleton B 
<->  B  =  { A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   class class class wbr 4023    _I cid 4304    X. cxp 4687  Singletoncsingle 24381
This theorem is referenced by:  elsingles  24457  fnsingle  24458  fvsingle  24459  brapply  24477  brsuccf  24480  funpartfun  24481  funpartfv  24483
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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