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Theorem qsss 6720
Description: A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
qsss.1  |-  ( ph  ->  R  Er  A )
Assertion
Ref Expression
qsss  |-  ( ph  ->  ( A /. R
)  C_  ~P A
)

Proof of Theorem qsss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . 4  |-  x  e. 
_V
21elqs 6712 . . 3  |-  ( x  e.  ( A /. R )  <->  E. y  e.  A  x  =  [ y ] R
)
3 qsss.1 . . . . . . 7  |-  ( ph  ->  R  Er  A )
43ecss 6701 . . . . . 6  |-  ( ph  ->  [ y ] R  C_  A )
5 sseq1 3199 . . . . . 6  |-  ( x  =  [ y ] R  ->  ( x  C_  A  <->  [ y ] R  C_  A ) )
64, 5syl5ibrcom 213 . . . . 5  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  C_  A ) )
71elpw 3631 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
86, 7syl6ibr 218 . . . 4  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  e.  ~P A
) )
98rexlimdvw 2670 . . 3  |-  ( ph  ->  ( E. y  e.  A  x  =  [
y ] R  ->  x  e.  ~P A
) )
102, 9syl5bi 208 . 2  |-  ( ph  ->  ( x  e.  ( A /. R )  ->  x  e.  ~P A ) )
1110ssrdv 3185 1  |-  ( ph  ->  ( A /. R
)  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625    Er wer 6657   [cec 6658   /.cqs 6659
This theorem is referenced by:  axcnex  8769  wuncn  8792  qshash  12285  lagsubg2  14678  lagsubg  14679  orbsta2  14768  sylow1lem3  14911  sylow2alem2  14929  sylow2a  14930  sylow2blem2  14932  sylow2blem3  14933  sylow3lem3  14940  sylow3lem4  14941  vitalilem5  18967  vitali  18968
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-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
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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-er 6660  df-ec 6662  df-qs 6666
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