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Theorem qsel 6754
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 2296 . . 3  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2357 . . . 4  |-  ( [ x ] R  =  B  ->  ( C  e.  [ x ] R  <->  C  e.  B ) )
3 eqeq1 2302 . . . 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 2804 . . . . . 6  |-  x  e. 
_V
6 elecg 6714 . . . . . 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 6722 . . . . 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 6742 . 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 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    Er wer 6673   [cec 6674   /.cqs 6675
This theorem is referenced by:  frgpnabllem2  15178  prter3  26853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-er 6676  df-ec 6678  df-qs 6682
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