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Theorem divslem 13461
Description: The function in divsval 13460 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
divsval.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
divsval.v  |-  ( ph  ->  V  =  ( Base `  R ) )
divsval.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
divsval.e  |-  ( ph  ->  .~  e.  W )
divsval.r  |-  ( ph  ->  R  e.  Z )
Assertion
Ref Expression
divslem  |-  ( 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 divslem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 divsval.e . . . . . 6  |-  ( ph  ->  .~  e.  W )
2 ecexg 6680 . . . . . 6  |-  (  .~  e.  W  ->  [ x ]  .~  e.  _V )
31, 2syl 15 . . . . 5  |-  ( ph  ->  [ x ]  .~  e.  _V )
43ralrimivw 2640 . . . 4  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
5 divsval.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
65fnmpt 5386 . . . 4  |-  ( A. x  e.  V  [
x ]  .~  e.  _V  ->  F  Fn  V
)
74, 6syl 15 . . 3  |-  ( ph  ->  F  Fn  V )
8 dffn4 5473 . . 3  |-  ( F  Fn  V  <->  F : V -onto-> ran  F )
97, 8sylib 188 . 2  |-  ( ph  ->  F : V -onto-> ran  F )
105rnmpt 4941 . . . 4  |-  ran  F  =  { y  |  E. x  e.  V  y  =  [ x ]  .~  }
11 df-qs 6682 . . . 4  |-  ( V /.  .~  )  =  { y  |  E. x  e.  V  y  =  [ x ]  .~  }
1210, 11eqtr4i 2319 . . 3  |-  ran  F  =  ( V /.  .~  )
13 foeq3 5465 . . 3  |-  ( ran 
F  =  ( V /.  .~  )  -> 
( F : V -onto-> ran  F  <->  F : V -onto-> ( V /.  .~  ) ) )
1412, 13ax-mp 8 . 2  |-  ( F : V -onto-> ran  F  <->  F : V -onto-> ( V /.  .~  ) )
159, 14sylib 188 1  |-  ( ph  ->  F : V -onto-> ( V /.  .~  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   [cec 6674   /.cqs 6675   Basecbs 13164    /.s cqus 13424
This theorem is referenced by:  divsbas  13463  divssca  13464  divsaddvallem  13469  divsaddflem  13470  divsaddval  13471  divsaddf  13472  divsmulval  13473  divsmulf  13474  divsgrp2  14629  divsrng2  15419  znzrhfo  16517  divstps  17429  divstgpopn  17818  divstgplem  17819  divstgphaus  17821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-fo 5277  df-ec 6678  df-qs 6682
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