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Theorem iunid 4366
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 4011 . . . . 5  |-  { x }  =  { y  |  y  =  x }
2 equcom 1778 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
32abbii 2575 . . . . 5  |-  { y  |  y  =  x }  =  { y  |  x  =  y }
41, 3eqtri 2470 . . . 4  |-  { x }  =  { y  |  x  =  y }
54a1i 11 . . 3  |-  ( x  e.  A  ->  { x }  =  { y  |  x  =  y } )
65iuneq2i 4330 . 2  |-  U_ x  e.  A  { x }  =  U_ x  e.  A  { y  |  x  =  y }
7 iunab 4357 . . 3  |-  U_ x  e.  A  { y  |  x  =  y }  =  { y  |  E. x  e.  A  x  =  y }
8 risset 2966 . . . 4  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
98abbii 2575 . . 3  |-  { y  |  y  e.  A }  =  { y  |  E. x  e.  A  x  =  y }
10 abid2 2581 . . 3  |-  { y  |  y  e.  A }  =  A
117, 9, 103eqtr2i 2476 . 2  |-  U_ x  e.  A  { y  |  x  =  y }  =  A
126, 11eqtri 2470 1  |-  U_ x  e.  A  { x }  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1381    e. wcel 1802   {cab 2426   E.wrex 2792   {csn 4010   U_ciun 4311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-rex 2797  df-v 3095  df-in 3465  df-ss 3472  df-sn 4011  df-iun 4313
This theorem is referenced by:  iunxpconst  5042  fvn0ssdmfun  6003  xpexgALT  6774  uniqs  7369  rankcf  9153  dprd2da  16959  t1ficld  19694  discmp  19764  xkoinjcn  20054  metnrmlem2  21230  ovoliunlem1  21779  i1fima  21951  i1fd  21954  itg1addlem5  21973  sibfof  28148  cvmlift2lem12  28625  dftrpred4g  29285  itg2addnclem2  30035  ftc1anclem6  30063  bnj1415  33801
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