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Theorem ecelqsg 6248
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 2083 . . 3  |-  [ B ] R  =  [ B ] R
2 eceq1 6230 . . . . 5  |-  ( x  =  B  ->  [ x ] R  =  [ B ] R )
32eqeq2d 2094 . . . 4  |-  ( x  =  B  ->  ( [ B ] R  =  [ x ] R  <->  [ B ] R  =  [ B ] R
) )
43rspcev 2711 . . 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 6199 . . . 4  |-  ( R  e.  V  ->  [ B ] R  e.  _V )
7 elqsg 6245 . . . 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 1285    e. wcel 1434   E.wrex 2354   _Vcvv 2611   [cec 6193   /.cqs 6194
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-xp 4398  df-cnv 4400  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-ec 6197  df-qs 6201
This theorem is referenced by:  ecelqsi  6249  qliftlem  6273  eroprf  6288
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