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Theorem divslem 13770
Description: The function in divsval 13769 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 6911 . . . . . 6  |-  (  .~  e.  W  ->  [ x ]  .~  e.  _V )
31, 2syl 16 . . . . 5  |-  ( ph  ->  [ x ]  .~  e.  _V )
43ralrimivw 2792 . . . 4  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
5 divsval.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
65fnmpt 5573 . . . 4  |-  ( A. x  e.  V  [
x ]  .~  e.  _V  ->  F  Fn  V
)
74, 6syl 16 . . 3  |-  ( ph  ->  F  Fn  V )
8 dffn4 5661 . . 3  |-  ( F  Fn  V  <->  F : V -onto-> ran  F )
97, 8sylib 190 . 2  |-  ( ph  ->  F : V -onto-> ran  F )
105rnmpt 5118 . . . 4  |-  ran  F  =  { y  |  E. x  e.  V  y  =  [ x ]  .~  }
11 df-qs 6913 . . . 4  |-  ( V /.  .~  )  =  { y  |  E. x  e.  V  y  =  [ x ]  .~  }
1210, 11eqtr4i 2461 . . 3  |-  ran  F  =  ( V /.  .~  )
13 foeq3 5653 . . 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 190 1  |-  ( ph  ->  F : V -onto-> ( V /.  .~  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   _Vcvv 2958    e. cmpt 4268   ran crn 4881    Fn wfn 5451   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083   [cec 6905   /.cqs 6906   Basecbs 13471    /.s cqus 13733
This theorem is referenced by:  divsbas  13772  divssca  13773  divsaddvallem  13778  divsaddflem  13779  divsaddval  13780  divsaddf  13781  divsmulval  13782  divsmulf  13783  divsgrp2  14938  divsrng2  15728  znzrhfo  16830  divstps  17756  divstgpopn  18151  divstgplem  18152  divstgphaus  18154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5458  df-fn 5459  df-fo 5462  df-ec 6909  df-qs 6913
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