MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ecelqsdm Unicode version

Theorem ecelqsdm 6941
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
StepHypRef Expression
1 elqsn0 6940 . . 3  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  [ B ] R  =/=  (/) )
2 ecdmn0 6914 . . 3  |-  ( B  e.  dom  R  <->  [ B ] R  =/=  (/) )
31, 2sylibr 204 . 2  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  B  e.  dom  R )
4 simpl 444 . 2  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  dom  R  =  A )
53, 4eleqtrd 2488 1  |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   (/)c0 3596   dom cdm 4845   [cec 6870   /.cqs 6871
This theorem is referenced by:  brecop2  6965  th3qlem1  6977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ec 6874  df-qs 6878
  Copyright terms: Public domain W3C validator