ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecelqsdm Unicode version

Theorem ecelqsdm 6499
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 6497 . . 3  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  E. x  x  e.  [ B ] R )
2 ecdmn0m 6471 . . 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 2218 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 1331   E.wex 1468    e. wcel 1480   dom cdm 4539   [cec 6427   /.cqs 6428
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431  df-qs 6435
This theorem is referenced by:  th3qlem1  6531  nnnq0lem1  7266  prsrlem1  7562  gt0srpr  7568
  Copyright terms: Public domain W3C validator