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Theorem iunid 3768
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 3437 . . . . 5  |-  { x }  =  { y  |  y  =  x }
2 equcom 1637 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
32abbii 2200 . . . . 5  |-  { y  |  y  =  x }  =  { y  |  x  =  y }
41, 3eqtri 2105 . . . 4  |-  { x }  =  { y  |  x  =  y }
54a1i 9 . . 3  |-  ( x  e.  A  ->  { x }  =  { y  |  x  =  y } )
65iuneq2i 3731 . 2  |-  U_ x  e.  A  { x }  =  U_ x  e.  A  { y  |  x  =  y }
7 iunab 3759 . . 3  |-  U_ x  e.  A  { y  |  x  =  y }  =  { y  |  E. x  e.  A  x  =  y }
8 risset 2402 . . . 4  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
98abbii 2200 . . 3  |-  { y  |  y  e.  A }  =  { y  |  E. x  e.  A  x  =  y }
10 abid2 2205 . . 3  |-  { y  |  y  e.  A }  =  A
117, 9, 103eqtr2i 2111 . 2  |-  U_ x  e.  A  { y  |  x  =  y }  =  A
126, 11eqtri 2105 1  |-  U_ x  e.  A  { x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1287    e. wcel 1436   {cab 2071   E.wrex 2356   {csn 3431   U_ciun 3713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-in 2994  df-ss 3001  df-sn 3437  df-iun 3715
This theorem is referenced by:  iunxpconst  4466  xpexgALT  5861  uniqs  6302
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