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Theorem qsss 7160
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 2974 . . . 4  |-  x  e. 
_V
21elqs 7152 . . 3  |-  ( x  e.  ( A /. R )  <->  E. y  e.  A  x  =  [ y ] R
)
3 qsss.1 . . . . . . 7  |-  ( ph  ->  R  Er  A )
43ecss 7141 . . . . . 6  |-  ( ph  ->  [ y ] R  C_  A )
5 sseq1 3376 . . . . . 6  |-  ( x  =  [ y ] R  ->  ( x  C_  A  <->  [ y ] R  C_  A ) )
64, 5syl5ibrcom 222 . . . . 5  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  C_  A ) )
7 selpw 3866 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
86, 7syl6ibr 227 . . . 4  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  e.  ~P A
) )
98rexlimdvw 2843 . . 3  |-  ( ph  ->  ( E. y  e.  A  x  =  [
y ] R  ->  x  e.  ~P A
) )
102, 9syl5bi 217 . 2  |-  ( ph  ->  ( x  e.  ( A /. R )  ->  x  e.  ~P A ) )
1110ssrdv 3361 1  |-  ( ph  ->  ( A /. R
)  C_  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   E.wrex 2715    C_ wss 3327   ~Pcpw 3859    Er wer 7097   [cec 7098   /.cqs 7099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-er 7100  df-ec 7102  df-qs 7106
This theorem is referenced by:  axcnex  9313  wuncn  9336  qshash  13289  lagsubg2  15741  lagsubg  15742  orbsta2  15831  sylow1lem3  16098  sylow2alem2  16116  sylow2a  16117  sylow2blem2  16119  sylow2blem3  16120  sylow3lem3  16127  sylow3lem4  16128  vitalilem5  21091  vitali  21092  qerclwwlknfi  30501
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