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Theorem iunid 3838
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 3503 . . . . 5  |-  { x }  =  { y  |  y  =  x }
2 equcom 1667 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
32abbii 2233 . . . . 5  |-  { y  |  y  =  x }  =  { y  |  x  =  y }
41, 3eqtri 2138 . . . 4  |-  { x }  =  { y  |  x  =  y }
54a1i 9 . . 3  |-  ( x  e.  A  ->  { x }  =  { y  |  x  =  y } )
65iuneq2i 3801 . 2  |-  U_ x  e.  A  { x }  =  U_ x  e.  A  { y  |  x  =  y }
7 iunab 3829 . . 3  |-  U_ x  e.  A  { y  |  x  =  y }  =  { y  |  E. x  e.  A  x  =  y }
8 risset 2440 . . . 4  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
98abbii 2233 . . 3  |-  { y  |  y  e.  A }  =  { y  |  E. x  e.  A  x  =  y }
10 abid2 2238 . . 3  |-  { y  |  y  e.  A }  =  A
117, 9, 103eqtr2i 2144 . 2  |-  U_ x  e.  A  { y  |  x  =  y }  =  A
126, 11eqtri 2138 1  |-  U_ x  e.  A  { x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394   {csn 3497   U_ciun 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-sn 3503  df-iun 3785
This theorem is referenced by:  abnexg  4337  iunxpconst  4569  xpexgALT  5999  uniqs  6455
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