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Theorem uniqs 7328
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 7272 . . . . 5  |-  ( R  e.  V  ->  [ x ] R  e.  _V )
21ralrimivw 2818 . . . 4  |-  ( R  e.  V  ->  A. x  e.  A  [ x ] R  e.  _V )
3 dfiun2g 4302 . . . 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 17 . . 3  |-  ( R  e.  V  ->  U_ x  e.  A  [ x ] R  =  U. { y  |  E. x  e.  A  y  =  [ x ] R } )
54eqcomd 2410 . 2  |-  ( R  e.  V  ->  U. {
y  |  E. x  e.  A  y  =  [ x ] R }  =  U_ x  e.  A  [ x ] R )
6 df-qs 7274 . . 3  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
76unieqi 4199 . 2  |-  U. ( A /. R )  = 
U. { y  |  E. x  e.  A  y  =  [ x ] R }
8 df-ec 7270 . . . . 5  |-  [ x ] R  =  ( R " { x }
)
98a1i 11 . . . 4  |-  ( x  e.  A  ->  [ x ] R  =  ( R " { x }
) )
109iuneq2i 4289 . . 3  |-  U_ x  e.  A  [ x ] R  =  U_ x  e.  A  ( R " { x }
)
11 imaiun 6094 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  U_ x  e.  A  ( R " { x } )
12 iunid 4325 . . . 4  |-  U_ x  e.  A  { x }  =  A
1312imaeq2i 5276 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  ( R " A )
1410, 11, 133eqtr2ri 2438 . 2  |-  ( R
" A )  = 
U_ x  e.  A  [ x ] R
155, 7, 143eqtr4g 2468 1  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   {cab 2387   A.wral 2753   E.wrex 2754   _Vcvv 3058   {csn 3971   U.cuni 4190   U_ciun 4270   "cima 4945   [cec 7266   /.cqs 7267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-xp 4948  df-cnv 4950  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-ec 7270  df-qs 7274
This theorem is referenced by:  uniqs2  7330  ecqs  7332
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