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Theorem elqsn0m 6501
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 2140 . 2  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2204 . . 3  |-  ( [ y ] R  =  B  ->  ( x  e.  [ y ] R  <->  x  e.  B ) )
32exbidv 1798 . 2  |-  ( [ y ] R  =  B  ->  ( E. x  x  e.  [ y ] R  <->  E. x  x  e.  B )
)
4 eleq2 2204 . . . 4  |-  ( dom 
R  =  A  -> 
( y  e.  dom  R  <-> 
y  e.  A ) )
54biimpar 295 . . 3  |-  ( ( dom  R  =  A  /\  y  e.  A
)  ->  y  e.  dom  R )
6 ecdmn0m 6475 . . 3  |-  ( y  e.  dom  R  <->  E. x  x  e.  [ y ] R )
75, 6sylib 121 . 2  |-  ( ( dom  R  =  A  /\  y  e.  A
)  ->  E. x  x  e.  [ y ] R )
81, 3, 7ectocld 6499 1  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  E. x  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332   E.wex 1469    e. wcel 1481   dom cdm 4543   [cec 6431   /.cqs 6432
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-xp 4549  df-cnv 4551  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-ec 6435  df-qs 6439
This theorem is referenced by:  elqsn0  6502  ecelqsdm  6503
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