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Theorem elqsn0m 6629
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 2189 . 2  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2253 . . 3  |-  ( [ y ] R  =  B  ->  ( x  e.  [ y ] R  <->  x  e.  B ) )
32exbidv 1836 . 2  |-  ( [ y ] R  =  B  ->  ( E. x  x  e.  [ y ] R  <->  E. x  x  e.  B )
)
4 eleq2 2253 . . . 4  |-  ( dom 
R  =  A  -> 
( y  e.  dom  R  <-> 
y  e.  A ) )
54biimpar 297 . . 3  |-  ( ( dom  R  =  A  /\  y  e.  A
)  ->  y  e.  dom  R )
6 ecdmn0m 6603 . . 3  |-  ( y  e.  dom  R  <->  E. x  x  e.  [ y ] R )
75, 6sylib 122 . 2  |-  ( ( dom  R  =  A  /\  y  e.  A
)  ->  E. x  x  e.  [ y ] R )
81, 3, 7ectocld 6627 1  |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  E. x  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364   E.wex 1503    e. wcel 2160   dom cdm 4644   [cec 6557   /.cqs 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-ec 6561  df-qs 6565
This theorem is referenced by:  elqsn0  6630  ecelqsdm  6631
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