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Theorem qsel 7408
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 2457 . . 3  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2530 . . . 4  |-  ( [ x ] R  =  B  ->  ( C  e.  [ x ] R  <->  C  e.  B ) )
3 eqeq1 2461 . . . 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 3112 . . . . . 6  |-  x  e. 
_V
6 elecg 7368 . . . . . 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 7376 . . . . 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 7396 . 2  |-  ( ( R  Er  X  /\  B  e.  ( A /. R ) )  -> 
( C  e.  B  ->  B  =  [ C ] R ) )
15143impia 1193 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   class class class wbr 4456    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-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  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-co 5017  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:  frgpnabllem2  17005  prter3  30828
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