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Theorem qsel 7179
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 2443 . . 3  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2504 . . . 4  |-  ( [ x ] R  =  B  ->  ( C  e.  [ x ] R  <->  C  e.  B ) )
3 eqeq1 2449 . . . 4  |-  ( [ x ] R  =  B  ->  ( [
x ] R  =  [ C ] R  <->  B  =  [ C ] R ) )
42, 3imbi12d 320 . . 3  |-  ( [ x ] R  =  B  ->  ( ( C  e.  [ x ] R  ->  [ x ] R  =  [ C ] R )  <->  ( C  e.  B  ->  B  =  [ C ] R
) ) )
5 vex 2975 . . . . . 6  |-  x  e. 
_V
6 elecg 7139 . . . . . 6  |-  ( ( C  e.  [ x ] R  /\  x  e.  _V )  ->  ( C  e.  [ x ] R  <->  x R C ) )
75, 6mpan2 671 . . . . 5  |-  ( C  e.  [ x ] R  ->  ( C  e. 
[ x ] R  <->  x R C ) )
87ibi 241 . . . 4  |-  ( C  e.  [ x ] R  ->  x R C )
9 simpll 753 . . . . . 6  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  R  Er  X )
10 simpr 461 . . . . . 6  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  x R C )
119, 10erthi 7147 . . . . 5  |-  ( ( ( R  Er  X  /\  x  e.  A
)  /\  x R C )  ->  [ x ] R  =  [ C ] R )
1211ex 434 . . . 4  |-  ( ( R  Er  X  /\  x  e.  A )  ->  ( x R C  ->  [ x ] R  =  [ C ] R ) )
138, 12syl5 32 . . 3  |-  ( ( R  Er  X  /\  x  e.  A )  ->  ( C  e.  [
x ] R  ->  [ x ] R  =  [ C ] R
) )
141, 4, 13ectocld 7167 . 2  |-  ( ( R  Er  X  /\  B  e.  ( A /. R ) )  -> 
( C  e.  B  ->  B  =  [ C ] R ) )
15143impia 1184 1  |-  ( ( R  Er  X  /\  B  e.  ( A /. R )  /\  C  e.  B )  ->  B  =  [ C ] R
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972   class class class wbr 4292    Er wer 7098   [cec 7099   /.cqs 7100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-er 7101  df-ec 7103  df-qs 7107
This theorem is referenced by:  frgpnabllem2  16352  prter3  29027
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