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Theorem qsel 6738
Description: If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
qsel  |-  ( ( R  Er  X  /\  B  e.  ( A /. R )  /\  C  e.  B )  ->  B  =  [ C ] R
)

Proof of Theorem qsel
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2344 . . . 4  |-  ( [ x ] R  =  B  ->  ( C  e.  [ x ] R  <->  C  e.  B ) )
3 eqeq1 2289 . . . 4  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =  [ C ] R  <->  B  =  [ C ] R ) )
42, 3imbi12d 311 . . 3  |-  ( [ x ] R  =  B  ->  ( ( C  e.  [ x ] R  ->  [ x ] R  =  [ C ] R )  <->  ( C  e.  B  ->  B  =  [ C ] R
) ) )
5 vex 2791 . . . . . 6  |-  x  e. 
_V
6 elecg 6698 . . . . . 6  |-  ( ( C  e.  [ x ] R  /\  x  e.  _V )  ->  ( C  e.  [ x ] R  <->  x R C ) )
75, 6mpan2 652 . . . . 5  |-  ( C  e.  [ x ] R  ->  ( C  e. 
[ x ] R  <->  x R C ) )
87ibi 232 . . . 4  |-  ( C  e.  [ x ] R  ->  x R C )
9 simpll 730 . . . . . 6  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  R  Er  X )
10 simpr 447 . . . . . 6  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  x R C )
119, 10erthi 6706 . . . . 5  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  [ x ] R  =  [ C ] R )
1211ex 423 . . . 4  |-  ( ( R  Er  X  /\  x  e.  A )  ->  ( x R C  ->  [ x ] R  =  [ C ] R ) )
138, 12syl5 28 . . 3  |-  ( ( R  Er  X  /\  x  e.  A )  ->  ( C  e.  [
x ] R  ->  [ x ] R  =  [ C ] R
) )
141, 4, 13ectocld 6726 . 2  |-  ( ( R  Er  X  /\  B  e.  ( A /. R ) )  -> 
( C  e.  B  ->  B  =  [ C ] R ) )
15143impia 1148 1  |-  ( ( R  Er  X  /\  B  e.  ( A /. R )  /\  C  e.  B )  ->  B  =  [ C ] R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    Er wer 6657   [cec 6658   /.cqs 6659
This theorem is referenced by:  frgpnabllem2  15162  prter3  26750
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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  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-co 4698  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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