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Theorem quslem 12804
Description: The function in qusval 12803 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
qusval.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
qusval.v  |-  ( ph  ->  V  =  ( Base `  R ) )
qusval.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
qusval.e  |-  ( ph  ->  .~  e.  W )
qusval.r  |-  ( ph  ->  R  e.  Z )
Assertion
Ref Expression
quslem  |-  ( ph  ->  F : V -onto-> ( V /.  .~  ) )
Distinct variable groups:    x,  .~    ph, x    x, R    x, V
Allowed substitution hints:    U( x)    F( x)    W( x)    Z( x)

Proof of Theorem quslem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 qusval.e . . . . . 6  |-  ( ph  ->  .~  e.  W )
2 ecexg 6564 . . . . . 6  |-  (  .~  e.  W  ->  [ x ]  .~  e.  _V )
31, 2syl 14 . . . . 5  |-  ( ph  ->  [ x ]  .~  e.  _V )
43ralrimivw 2564 . . . 4  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
5 qusval.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
65fnmpt 5361 . . . 4  |-  ( A. x  e.  V  [
x ]  .~  e.  _V  ->  F  Fn  V
)
74, 6syl 14 . . 3  |-  ( ph  ->  F  Fn  V )
8 dffn4 5463 . . 3  |-  ( F  Fn  V  <->  F : V -onto-> ran  F )
97, 8sylib 122 . 2  |-  ( ph  ->  F : V -onto-> ran  F )
105rnmpt 4893 . . . 4  |-  ran  F  =  { y  |  E. x  e.  V  y  =  [ x ]  .~  }
11 df-qs 6566 . . . 4  |-  ( V /.  .~  )  =  { y  |  E. x  e.  V  y  =  [ x ]  .~  }
1210, 11eqtr4i 2213 . . 3  |-  ran  F  =  ( V /.  .~  )
13 foeq3 5455 . . 3  |-  ( ran 
F  =  ( V /.  .~  )  -> 
( F : V -onto-> ran  F  <->  F : V -onto-> ( V /.  .~  ) ) )
1412, 13ax-mp 5 . 2  |-  ( F : V -onto-> ran  F  <->  F : V -onto-> ( V /.  .~  ) )
159, 14sylib 122 1  |-  ( ph  ->  F : V -onto-> ( V /.  .~  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   _Vcvv 2752    |-> cmpt 4079   ran crn 4645    Fn wfn 5230   -onto->wfo 5233   ` cfv 5235  (class class class)co 5897   [cec 6558   /.cqs 6559   Basecbs 12515    /.s cqus 12780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-fun 5237  df-fn 5238  df-fo 5241  df-ec 6562  df-qs 6566
This theorem is referenced by:  qusbas  12807  qusaddvallemg  12812  qusaddflemg  12813  qusaddval  12814  qusaddf  12815  qusmulval  12816  qusmulf  12817  qusgrp2  13070  qusrng  13329  qusring2  13433
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