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Theorem iunid 3868
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 3533 . . . . 5  |-  { x }  =  { y  |  y  =  x }
2 equcom 1682 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
32abbii 2255 . . . . 5  |-  { y  |  y  =  x }  =  { y  |  x  =  y }
41, 3eqtri 2160 . . . 4  |-  { x }  =  { y  |  x  =  y }
54a1i 9 . . 3  |-  ( x  e.  A  ->  { x }  =  { y  |  x  =  y } )
65iuneq2i 3831 . 2  |-  U_ x  e.  A  { x }  =  U_ x  e.  A  { y  |  x  =  y }
7 iunab 3859 . . 3  |-  U_ x  e.  A  { y  |  x  =  y }  =  { y  |  E. x  e.  A  x  =  y }
8 risset 2463 . . . 4  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
98abbii 2255 . . 3  |-  { y  |  y  e.  A }  =  { y  |  E. x  e.  A  x  =  y }
10 abid2 2260 . . 3  |-  { y  |  y  e.  A }  =  A
117, 9, 103eqtr2i 2166 . 2  |-  U_ x  e.  A  { y  |  x  =  y }  =  A
126, 11eqtri 2160 1  |-  U_ x  e.  A  { x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480   {cab 2125   E.wrex 2417   {csn 3527   U_ciun 3813
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533  df-iun 3815
This theorem is referenced by:  abnexg  4367  iunxpconst  4599  xpexgALT  6031  uniqs  6487
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