ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uniqs Unicode version

Theorem uniqs 6493
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 6439 . . . . 5  |-  ( R  e.  V  ->  [ x ] R  e.  _V )
21ralrimivw 2509 . . . 4  |-  ( R  e.  V  ->  A. x  e.  A  [ x ] R  e.  _V )
3 dfiun2g 3851 . . . 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 2146 . 2  |-  ( R  e.  V  ->  U. {
y  |  E. x  e.  A  y  =  [ x ] R }  =  U_ x  e.  A  [ x ] R )
6 df-qs 6441 . . 3  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
76unieqi 3752 . 2  |-  U. ( A /. R )  = 
U. { y  |  E. x  e.  A  y  =  [ x ] R }
8 df-ec 6437 . . . . 5  |-  [ x ] R  =  ( R " { x }
)
98a1i 9 . . . 4  |-  ( x  e.  A  ->  [ x ] R  =  ( R " { x }
) )
109iuneq2i 3837 . . 3  |-  U_ x  e.  A  [ x ] R  =  U_ x  e.  A  ( R " { x }
)
11 imaiun 5667 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  U_ x  e.  A  ( R " { x } )
12 iunid 3874 . . . 4  |-  U_ x  e.  A  { x }  =  A
1312imaeq2i 4885 . . 3  |-  ( R
" U_ x  e.  A  { x } )  =  ( R " A )
1410, 11, 133eqtr2ri 2168 . 2  |-  ( R
" A )  = 
U_ x  e.  A  [ x ] R
155, 7, 143eqtr4g 2198 1  |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   _Vcvv 2689   {csn 3530   U.cuni 3742   U_ciun 3819   "cima 4548   [cec 6433   /.cqs 6434
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-iun 3821  df-br 3936  df-opab 3996  df-xp 4551  df-cnv 4553  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-ec 6437  df-qs 6441
This theorem is referenced by:  uniqs2  6495  ecqs  6497
  Copyright terms: Public domain W3C validator