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Theorem elqsn0m 6240
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 2082 . 2  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2143 . . 3  |-  ( [ y ] R  =  B  ->  ( x  e.  [ y ] R  <->  x  e.  B ) )
32exbidv 1747 . 2  |-  ( [ y ] R  =  B  ->  ( E. x  x  e.  [ y ] R  <->  E. x  x  e.  B )
)
4 eleq2 2143 . . . 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 6214 . . 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 6238 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 1285   E.wex 1422    e. wcel 1434   dom cdm 4371   [cec 6170   /.cqs 6171
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-ec 6174  df-qs 6178
This theorem is referenced by:  elqsn0  6241  ecelqsdm  6242
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