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Theorem uniqs 7368
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 7312 . . . . 5  |-  ( R  e.  V  ->  [ x ] R  e.  _V )
21ralrimivw 2879 . . . 4  |-  ( R  e.  V  ->  A. x  e.  A  [ x ] R  e.  _V )
3 dfiun2g 4357 . . . 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 16 . . 3  |-  ( R  e.  V  ->  U_ x  e.  A  [ x ] R  =  U. { y  |  E. x  e.  A  y  =  [ x ] R } )
54eqcomd 2475 . 2  |-  ( R  e.  V  ->  U. {
y  |  E. x  e.  A  y  =  [ x ] R }  =  U_ x  e.  A  [ x ] R )
6 df-qs 7314 . . 3  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
76unieqi 4254 . 2  |-  U. ( A /. R )  = 
U. { y  |  E. x  e.  A  y  =  [ x ] R }
8 df-ec 7310 . . . . 5  |-  [ x ] R  =  ( R " { x }
)
98a1i 11 . . . 4  |-  ( x  e.  A  ->  [ x ] R  =  ( R " { x }
) )
109iuneq2i 4344 . . 3  |-  U_ x  e.  A  [ x ] R  =  U_ x  e.  A  ( R " { x }
)
11 imaiun 6143 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  U_ x  e.  A  ( R " { x } )
12 iunid 4380 . . . 4  |-  U_ x  e.  A  { x }  =  A
1312imaeq2i 5333 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  ( R " A )
1410, 11, 133eqtr2ri 2503 . 2  |-  ( R
" A )  = 
U_ x  e.  A  [ x ] R
155, 7, 143eqtr4g 2533 1  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113   {csn 4027   U.cuni 4245   U_ciun 4325   "cima 5002   [cec 7306   /.cqs 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-ec 7310  df-qs 7314
This theorem is referenced by:  uniqs2  7370  ecqs  7372
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