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Theorem divslem 13445
Description: The function in divsval 13444 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 6664 . . . . . 6  |-  (  .~  e.  W  ->  [ x ]  .~  e.  _V )
31, 2syl 15 . . . . 5  |-  ( ph  ->  [ x ]  .~  e.  _V )
43ralrimivw 2627 . . . 4  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
5 divsval.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
65fnmpt 5370 . . . 4  |-  ( A. x  e.  V  [
x ]  .~  e.  _V  ->  F  Fn  V
)
74, 6syl 15 . . 3  |-  ( ph  ->  F  Fn  V )
8 dffn4 5457 . . 3  |-  ( F  Fn  V  <->  F : V -onto-> ran  F )
97, 8sylib 188 . 2  |-  ( ph  ->  F : V -onto-> ran  F )
105rnmpt 4925 . . . 4  |-  ran  F  =  { y  |  E. x  e.  V  y  =  [ x ]  .~  }
11 df-qs 6666 . . . 4  |-  ( V /.  .~  )  =  { y  |  E. x  e.  V  y  =  [ x ]  .~  }
1210, 11eqtr4i 2306 . . 3  |-  ran  F  =  ( V /.  .~  )
13 foeq3 5449 . . 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 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    e. cmpt 4077   ran crn 4690    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   [cec 6658   /.cqs 6659   Basecbs 13148    /.s cqus 13408
This theorem is referenced by:  divsbas  13447  divssca  13448  divsaddvallem  13453  divsaddflem  13454  divsaddval  13455  divsaddf  13456  divsmulval  13457  divsmulf  13458  divsgrp2  14613  divsrng2  15403  znzrhfo  16501  divstps  17413  divstgpopn  17802  divstgplem  17803  divstgphaus  17805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-fo 5261  df-ec 6662  df-qs 6666
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