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Theorem elqsn0m 6579
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 2170 . 2  |-  ( A /. R )  =  ( A /. R
)
2 eleq2 2234 . . 3  |-  ( [ y ] R  =  B  ->  ( x  e.  [ y ] R  <->  x  e.  B ) )
32exbidv 1818 . 2  |-  ( [ y ] R  =  B  ->  ( E. x  x  e.  [ y ] R  <->  E. x  x  e.  B )
)
4 eleq2 2234 . . . 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 6553 . . 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 6577 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 1348   E.wex 1485    e. wcel 2141   dom cdm 4609   [cec 6509   /.cqs 6510
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-cnv 4617  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-ec 6513  df-qs 6517
This theorem is referenced by:  elqsn0  6580  ecelqsdm  6581
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