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Theorem elqsn0m 6340
Description: An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
Assertion
Ref Expression
elqsn0m  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  E. x  x  e.  B )
Distinct variable groups:    x, R    x, A    x, B

Proof of Theorem elqsn0m
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2088 . 2  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2151 . . 3  |-  ( [ y ] R  =  B  ->  ( x  e.  [ y ] R  <->  x  e.  B ) )
32exbidv 1753 . 2  |-  ( [ y ] R  =  B  ->  ( E. x  x  e.  [ y ] R  <->  E. x  x  e.  B )
)
4 eleq2 2151 . . . 4  |-  ( dom 
R  =  A  -> 
( y  e.  dom  R  <-> 
y  e.  A ) )
54biimpar 291 . . 3  |-  ( ( dom  R  =  A  /\  y  e.  A
)  ->  y  e.  dom  R )
6 ecdmn0m 6314 . . 3  |-  ( y  e.  dom  R  <->  E. x  x  e.  [ y ] R )
75, 6sylib 120 . 2  |-  ( ( dom  R  =  A  /\  y  e.  A
)  ->  E. x  x  e.  [ y ] R )
81, 3, 7ectocld 6338 1  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  E. x  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   dom cdm 4428   [cec 6270   /.cqs 6271
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-ec 6274  df-qs 6278
This theorem is referenced by:  elqsn0  6341  ecelqsdm  6342
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