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Theorem iunid 3928
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 3589 . . . . 5  |-  { x }  =  { y  |  y  =  x }
2 equcom 1699 . . . . . 6  |-  ( y  =  x  <->  x  =  y )
32abbii 2286 . . . . 5  |-  { y  |  y  =  x }  =  { y  |  x  =  y }
41, 3eqtri 2191 . . . 4  |-  { x }  =  { y  |  x  =  y }
54a1i 9 . . 3  |-  ( x  e.  A  ->  { x }  =  { y  |  x  =  y } )
65iuneq2i 3891 . 2  |-  U_ x  e.  A  { x }  =  U_ x  e.  A  { y  |  x  =  y }
7 iunab 3919 . . 3  |-  U_ x  e.  A  { y  |  x  =  y }  =  { y  |  E. x  e.  A  x  =  y }
8 risset 2498 . . . 4  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
98abbii 2286 . . 3  |-  { y  |  y  e.  A }  =  { y  |  E. x  e.  A  x  =  y }
10 abid2 2291 . . 3  |-  { y  |  y  e.  A }  =  A
117, 9, 103eqtr2i 2197 . 2  |-  U_ x  e.  A  { y  |  x  =  y }  =  A
126, 11eqtri 2191 1  |-  U_ x  e.  A  { x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141   {cab 2156   E.wrex 2449   {csn 3583   U_ciun 3873
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589  df-iun 3875
This theorem is referenced by:  abnexg  4431  iunxpconst  4671  xpexgALT  6112  uniqs  6571
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