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Theorem uniqs 6439
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 6385 . . . . 5  |-  ( R  e.  V  ->  [ x ] R  e.  _V )
21ralrimivw 2478 . . . 4  |-  ( R  e.  V  ->  A. x  e.  A  [ x ] R  e.  _V )
3 dfiun2g 3809 . . . 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 2118 . 2  |-  ( R  e.  V  ->  U. {
y  |  E. x  e.  A  y  =  [ x ] R }  =  U_ x  e.  A  [ x ] R )
6 df-qs 6387 . . 3  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
76unieqi 3710 . 2  |-  U. ( A /. R )  = 
U. { y  |  E. x  e.  A  y  =  [ x ] R }
8 df-ec 6383 . . . . 5  |-  [ x ] R  =  ( R " { x }
)
98a1i 9 . . . 4  |-  ( x  e.  A  ->  [ x ] R  =  ( R " { x }
) )
109iuneq2i 3795 . . 3  |-  U_ x  e.  A  [ x ] R  =  U_ x  e.  A  ( R " { x }
)
11 imaiun 5613 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  U_ x  e.  A  ( R " { x } )
12 iunid 3832 . . . 4  |-  U_ x  e.  A  { x }  =  A
1312imaeq2i 4835 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  ( R " A )
1410, 11, 133eqtr2ri 2140 . 2  |-  ( R
" A )  = 
U_ x  e.  A  [ x ] R
155, 7, 143eqtr4g 2170 1  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1312    e. wcel 1461   {cab 2099   A.wral 2388   E.wrex 2389   _Vcvv 2655   {csn 3491   U.cuni 3700   U_ciun 3777   "cima 4500   [cec 6379   /.cqs 6380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-xp 4503  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-ec 6383  df-qs 6387
This theorem is referenced by:  uniqs2  6441  ecqs  6443
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