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Theorem iunid 4325
Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
Assertion
Ref Expression
iunid  |-  U_ x  e.  A  { x }  =  A
Distinct variable group:    x, A

Proof of Theorem iunid
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-sn 3978 . . . . 5  |-  { x }  =  { y  |  y  =  x }
2 equcom 1734 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
32abbii 2585 . . . . 5  |-  { y  |  y  =  x }  =  { y  |  x  =  y }
41, 3eqtri 2480 . . . 4  |-  { x }  =  { y  |  x  =  y }
54a1i 11 . . 3  |-  ( x  e.  A  ->  { x }  =  { y  |  x  =  y } )
65iuneq2i 4289 . 2  |-  U_ x  e.  A  { x }  =  U_ x  e.  A  { y  |  x  =  y }
7 iunab 4316 . . 3  |-  U_ x  e.  A  { y  |  x  =  y }  =  { y  |  E. x  e.  A  x  =  y }
8 risset 2874 . . . 4  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
98abbii 2585 . . 3  |-  { y  |  y  e.  A }  =  { y  |  E. x  e.  A  x  =  y }
10 abid2 2591 . . 3  |-  { y  |  y  e.  A }  =  A
117, 9, 103eqtr2i 2486 . 2  |-  U_ x  e.  A  { y  |  x  =  y }  =  A
126, 11eqtri 2480 1  |-  U_ x  e.  A  { x }  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   {cab 2436   E.wrex 2796   {csn 3977   U_ciun 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-v 3072  df-in 3435  df-ss 3442  df-sn 3978  df-iun 4273
This theorem is referenced by:  iunxpconst  4995  xpexgALT  6672  uniqs  7262  rankcf  9047  dprd2da  16648  t1ficld  19049  discmp  19119  xkoinjcn  19378  metnrmlem2  20554  ovoliunlem1  21103  i1fima  21274  i1fd  21277  itg1addlem5  21296  sibfof  26862  cvmlift2lem12  27339  dftrpred4g  27834  itg2addnclem2  28584  ftc1anclem6  28612  bnj1415  32331
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