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Theorem iunid 3957
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 3613 . . . . 5  |-  { x }  =  { y  |  y  =  x }
2 equcom 1717 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
32abbii 2305 . . . . 5  |-  { y  |  y  =  x }  =  { y  |  x  =  y }
41, 3eqtri 2210 . . . 4  |-  { x }  =  { y  |  x  =  y }
54a1i 9 . . 3  |-  ( x  e.  A  ->  { x }  =  { y  |  x  =  y } )
65iuneq2i 3919 . 2  |-  U_ x  e.  A  { x }  =  U_ x  e.  A  { y  |  x  =  y }
7 iunab 3948 . . 3  |-  U_ x  e.  A  { y  |  x  =  y }  =  { y  |  E. x  e.  A  x  =  y }
8 risset 2518 . . . 4  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
98abbii 2305 . . 3  |-  { y  |  y  e.  A }  =  { y  |  E. x  e.  A  x  =  y }
10 abid2 2310 . . 3  |-  { y  |  y  e.  A }  =  A
117, 9, 103eqtr2i 2216 . 2  |-  U_ x  e.  A  { y  |  x  =  y }  =  A
126, 11eqtri 2210 1  |-  U_ x  e.  A  { x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160   {cab 2175   E.wrex 2469   {csn 3607   U_ciun 3901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-sn 3613  df-iun 3903
This theorem is referenced by:  abnexg  4464  iunxpconst  4704  xpexgALT  6159  uniqs  6620
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