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Theorem qsss 6165
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 2560 . . . 4  |-  x  e. 
_V
21elqs 6157 . . 3  |-  ( x  e.  ( A /. R )  <->  E. y  e.  A  x  =  [ y ] R
)
3 qsss.1 . . . . . . 7  |-  ( ph  ->  R  Er  A )
43ecss 6147 . . . . . 6  |-  ( ph  ->  [ y ] R  C_  A )
5 sseq1 2966 . . . . . 6  |-  ( x  =  [ y ] R  ->  ( x  C_  A  <->  [ y ] R  C_  A ) )
64, 5syl5ibrcom 146 . . . . 5  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  C_  A ) )
7 selpw 3366 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
86, 7syl6ibr 151 . . . 4  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  e.  ~P A
) )
98rexlimdvw 2436 . . 3  |-  ( ph  ->  ( E. y  e.  A  x  =  [
y ] R  ->  x  e.  ~P A
) )
102, 9syl5bi 141 . 2  |-  ( ph  ->  ( x  e.  ( A /. R )  ->  x  e.  ~P A ) )
1110ssrdv 2951 1  |-  ( ph  ->  ( A /. R
)  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   E.wrex 2307    C_ wss 2917   ~Pcpw 3359    Er wer 6103   [cec 6104   /.cqs 6105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-er 6106  df-ec 6108  df-qs 6112
This theorem is referenced by:  axcnex  6935
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