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Theorem brsingle 30223
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 30167 . 2  |- Singleton  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )  /_\  (  _I  (x)  _V ) ) )
4 brxp 4971 . . 3  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
51, 2, 4mpbir2an 919 . 2  |-  A ( _V  X.  _V ) B
6 elsn 3983 . . 3  |-  ( x  e.  { A }  <->  x  =  A )
71ideq 5095 . . 3  |-  ( x  _I  A  <->  x  =  A )
86, 7bitr4i 252 . 2  |-  ( x  e.  { A }  <->  x  _I  A )
91, 2, 3, 5, 8brtxpsd3 30202 1  |-  ( ASingleton B 
<->  B  =  { A } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1403    e. wcel 1840   _Vcvv 3056   {csn 3969   class class class wbr 4392    _I cid 4730    X. cxp 4938  Singletoncsingle 30143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-symdif 3667  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-eprel 4731  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fo 5529  df-fv 5531  df-1st 6736  df-2nd 6737  df-txp 30159  df-singleton 30167
This theorem is referenced by:  elsingles  30224  fnsingle  30225  fvsingle  30226  brapply  30244  brsuccf  30247  funpartlem  30248
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