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Theorem uniqs 6340
Description: The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
Assertion
Ref Expression
uniqs  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )

Proof of Theorem uniqs
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecexg 6286 . . . . 5  |-  ( R  e.  V  ->  [ x ] R  e.  _V )
21ralrimivw 2447 . . . 4  |-  ( R  e.  V  ->  A. x  e.  A  [ x ] R  e.  _V )
3 dfiun2g 3760 . . . 4  |-  ( A. x  e.  A  [
x ] R  e. 
_V  ->  U_ x  e.  A  [ x ] R  =  U. { y  |  E. x  e.  A  y  =  [ x ] R } )
42, 3syl 14 . . 3  |-  ( R  e.  V  ->  U_ x  e.  A  [ x ] R  =  U. { y  |  E. x  e.  A  y  =  [ x ] R } )
54eqcomd 2093 . 2  |-  ( R  e.  V  ->  U. {
y  |  E. x  e.  A  y  =  [ x ] R }  =  U_ x  e.  A  [ x ] R )
6 df-qs 6288 . . 3  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
76unieqi 3661 . 2  |-  U. ( A /. R )  = 
U. { y  |  E. x  e.  A  y  =  [ x ] R }
8 df-ec 6284 . . . . 5  |-  [ x ] R  =  ( R " { x }
)
98a1i 9 . . . 4  |-  ( x  e.  A  ->  [ x ] R  =  ( R " { x }
) )
109iuneq2i 3746 . . 3  |-  U_ x  e.  A  [ x ] R  =  U_ x  e.  A  ( R " { x }
)
11 imaiun 5531 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  U_ x  e.  A  ( R " { x } )
12 iunid 3783 . . . 4  |-  U_ x  e.  A  { x }  =  A
1312imaeq2i 4767 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  ( R " A )
1410, 11, 133eqtr2ri 2115 . 2  |-  ( R
" A )  = 
U_ x  e.  A  [ x ] R
155, 7, 143eqtr4g 2145 1  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619   {csn 3444   U.cuni 3651   U_ciun 3728   "cima 4439   [cec 6280   /.cqs 6281
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-iun 3730  df-br 3844  df-opab 3898  df-xp 4442  df-cnv 4444  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-ec 6284  df-qs 6288
This theorem is referenced by:  uniqs2  6342  ecqs  6344
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