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Theorem ecelqsg 6325
Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecelqsg  |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R
) )

Proof of Theorem ecelqsg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2088 . . 3  |-  [ B ] R  =  [ B ] R
2 eceq1 6307 . . . . 5  |-  ( x  =  B  ->  [ x ] R  =  [ B ] R )
32eqeq2d 2099 . . . 4  |-  ( x  =  B  ->  ( [ B ] R  =  [ x ] R  <->  [ B ] R  =  [ B ] R
) )
43rspcev 2722 . . 3  |-  ( ( B  e.  A  /\  [ B ] R  =  [ B ] R
)  ->  E. x  e.  A  [ B ] R  =  [
x ] R )
51, 4mpan2 416 . 2  |-  ( B  e.  A  ->  E. x  e.  A  [ B ] R  =  [
x ] R )
6 ecexg 6276 . . . 4  |-  ( R  e.  V  ->  [ B ] R  e.  _V )
7 elqsg 6322 . . . 4  |-  ( [ B ] R  e. 
_V  ->  ( [ B ] R  e.  ( A /. R )  <->  E. x  e.  A  [ B ] R  =  [
x ] R ) )
86, 7syl 14 . . 3  |-  ( R  e.  V  ->  ( [ B ] R  e.  ( A /. R
)  <->  E. x  e.  A  [ B ] R  =  [ x ] R
) )
98biimpar 291 . 2  |-  ( ( R  e.  V  /\  E. x  e.  A  [ B ] R  =  [
x ] R )  ->  [ B ] R  e.  ( A /. R ) )
105, 9sylan2 280 1  |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E.wrex 2360   _Vcvv 2619   [cec 6270   /.cqs 6271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-ec 6274  df-qs 6278
This theorem is referenced by:  ecelqsi  6326  qliftlem  6350  eroprf  6365
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