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Theorem iunid 3973
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 3659 . . . . 5  |-  { x }  =  { y  |  y  =  x }
2 equcom 1665 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
32abbii 2408 . . . . 5  |-  { y  |  y  =  x }  =  { y  |  x  =  y }
41, 3eqtri 2316 . . . 4  |-  { x }  =  { y  |  x  =  y }
54a1i 10 . . 3  |-  ( x  e.  A  ->  { x }  =  { y  |  x  =  y } )
65iuneq2i 3939 . 2  |-  U_ x  e.  A  { x }  =  U_ x  e.  A  { y  |  x  =  y }
7 iunab 3964 . . 3  |-  U_ x  e.  A  { y  |  x  =  y }  =  { y  |  E. x  e.  A  x  =  y }
8 risset 2603 . . . 4  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
98abbii 2408 . . 3  |-  { y  |  y  e.  A }  =  { y  |  E. x  e.  A  x  =  y }
10 abid2 2413 . . 3  |-  { y  |  y  e.  A }  =  A
117, 9, 103eqtr2i 2322 . 2  |-  U_ x  e.  A  { y  |  x  =  y }  =  A
126, 11eqtri 2316 1  |-  U_ x  e.  A  { x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   {csn 3653   U_ciun 3921
This theorem is referenced by:  iunxpconst  4762  xpexgALT  6086  uniqs  6735  rankcf  8415  dprd2da  15293  t1ficld  17071  discmp  17141  xkoinjcn  17397  metnrmlem2  18380  ovoliunlem1  18877  i1fima  19049  i1fd  19052  itg1addlem5  19071  cvmlift2lem12  23860  dftrpred4g  24308  itg2addnclem2  25004  bnj1415  29384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-sn 3659  df-iun 3923
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