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Theorem qsel 6912
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 2380 . . 3  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2441 . . . 4  |-  ( [ x ] R  =  B  ->  ( C  e.  [ x ] R  <->  C  e.  B ) )
3 eqeq1 2386 . . . 4  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =  [ C ] R  <->  B  =  [ C ] R ) )
42, 3imbi12d 312 . . 3  |-  ( [ x ] R  =  B  ->  ( ( C  e.  [ x ] R  ->  [ x ] R  =  [ C ] R )  <->  ( C  e.  B  ->  B  =  [ C ] R
) ) )
5 vex 2895 . . . . . 6  |-  x  e. 
_V
6 elecg 6872 . . . . . 6  |-  ( ( C  e.  [ x ] R  /\  x  e.  _V )  ->  ( C  e.  [ x ] R  <->  x R C ) )
75, 6mpan2 653 . . . . 5  |-  ( C  e.  [ x ] R  ->  ( C  e. 
[ x ] R  <->  x R C ) )
87ibi 233 . . . 4  |-  ( C  e.  [ x ] R  ->  x R C )
9 simpll 731 . . . . . 6  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  R  Er  X )
10 simpr 448 . . . . . 6  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  x R C )
119, 10erthi 6880 . . . . 5  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  [ x ] R  =  [ C ] R )
1211ex 424 . . . 4  |-  ( ( R  Er  X  /\  x  e.  A )  ->  ( x R C  ->  [ x ] R  =  [ C ] R ) )
138, 12syl5 30 . . 3  |-  ( ( R  Er  X  /\  x  e.  A )  ->  ( C  e.  [
x ] R  ->  [ x ] R  =  [ C ] R
) )
141, 4, 13ectocld 6900 . 2  |-  ( ( R  Er  X  /\  B  e.  ( A /. R ) )  -> 
( C  e.  B  ->  B  =  [ C ] R ) )
15143impia 1150 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2892   class class class wbr 4146    Er wer 6831   [cec 6832   /.cqs 6833
This theorem is referenced by:  frgpnabllem2  15405  prter3  26415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-er 6834  df-ec 6836  df-qs 6840
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