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Theorem ecelqsdm 6295
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 6293 . . 3  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  E. x  x  e.  [ B ] R )
2 ecdmn0m 6267 . . 3  |-  ( B  e.  dom  R  <->  E. x  x  e.  [ B ] R )
31, 2sylibr 132 . 2  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  B  e.  dom  R )
4 simpl 107 . 2  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  dom  R  =  A )
53, 4eleqtrd 2163 1  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287   E.wex 1424    e. wcel 1436   dom cdm 4404   [cec 6223   /.cqs 6224
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-cnv 4412  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-ec 6227  df-qs 6231
This theorem is referenced by:  th3qlem1  6327  nnnq0lem1  6926  prsrlem1  7209  gt0srpr  7215
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