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Theorem ecelqsdm 6467
Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
Assertion
Ref Expression
ecelqsdm  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  B  e.  A )

Proof of Theorem ecelqsdm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elqsn0m 6465 . . 3  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  E. x  x  e.  [ B ] R )
2 ecdmn0m 6439 . . 3  |-  ( B  e.  dom  R  <->  E. x  x  e.  [ B ] R )
31, 2sylibr 133 . 2  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  B  e.  dom  R )
4 simpl 108 . 2  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  dom  R  =  A )
53, 4eleqtrd 2196 1  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   dom cdm 4509   [cec 6395   /.cqs 6396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399  df-qs 6403
This theorem is referenced by:  th3qlem1  6499  nnnq0lem1  7222  prsrlem1  7518  gt0srpr  7524
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