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Theorem qsss 7390
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 3112 . . . 4  |-  x  e. 
_V
21elqs 7382 . . 3  |-  ( x  e.  ( A /. R )  <->  E. y  e.  A  x  =  [ y ] R
)
3 qsss.1 . . . . . . 7  |-  ( ph  ->  R  Er  A )
43ecss 7371 . . . . . 6  |-  ( ph  ->  [ y ] R  C_  A )
5 sseq1 3520 . . . . . 6  |-  ( x  =  [ y ] R  ->  ( x  C_  A  <->  [ y ] R  C_  A ) )
64, 5syl5ibrcom 222 . . . . 5  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  C_  A ) )
7 selpw 4022 . . . . 5  |-  ( x  e.  ~P A  <->  x  C_  A
)
86, 7syl6ibr 227 . . . 4  |-  ( ph  ->  ( x  =  [
y ] R  ->  x  e.  ~P A
) )
98rexlimdvw 2952 . . 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 3505 1  |-  ( ph  ->  ( A /. R
)  C_  ~P A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   E.wrex 2808    C_ wss 3471   ~Pcpw 4015    Er wer 7326   [cec 7327   /.cqs 7328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-er 7329  df-ec 7331  df-qs 7335
This theorem is referenced by:  axcnex  9541  wuncn  9564  qshash  13651  lagsubg2  16389  lagsubg  16390  orbsta2  16479  sylow1lem3  16747  sylow2alem2  16765  sylow2a  16766  sylow2blem2  16768  sylow2blem3  16769  sylow3lem3  16776  sylow3lem4  16777  vitalilem5  22147  vitali  22148  qerclwwlknfi  24956
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