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Theorem qsss 6596
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 2742 . . . 4  |-  x  e. 
_V
21elqs 6588 . . 3  |-  ( x  e.  ( A /. R )  <->  E. y  e.  A  x  =  [ y ] R
)
3 qsss.1 . . . . . . 7  |-  ( ph  ->  R  Er  A )
43ecss 6578 . . . . . 6  |-  ( ph  ->  [ y ] R  C_  A )
5 sseq1 3180 . . . . . 6  |-  ( x  =  [ y ] R  ->  ( x  C_  A  <->  [ y ] R  C_  A ) )
64, 5syl5ibrcom 157 . . . . 5  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  C_  A ) )
7 velpw 3584 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
86, 7imbitrrdi 162 . . . 4  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  e.  ~P A
) )
98rexlimdvw 2598 . . 3  |-  ( ph  ->  ( E. y  e.  A  x  =  [
y ] R  ->  x  e.  ~P A
) )
102, 9biimtrid 152 . 2  |-  ( ph  ->  ( x  e.  ( A /. R )  ->  x  e.  ~P A ) )
1110ssrdv 3163 1  |-  ( ph  ->  ( A /. R
)  C_  ~P A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   E.wrex 2456    C_ wss 3131   ~Pcpw 3577    Er wer 6534   [cec 6535   /.cqs 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-er 6537  df-ec 6539  df-qs 6543
This theorem is referenced by:  axcnex  7860
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