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Theorem ecdmn0m 6334
Description: A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
Assertion
Ref Expression
ecdmn0m  |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
Distinct variable groups:    x, R    x, A

Proof of Theorem ecdmn0m
StepHypRef Expression
1 elex 2630 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 ecexr 6297 . . 3  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
32exlimiv 1534 . 2  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
4 eldmg 4631 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
5 vex 2622 . . . . 5  |-  x  e. 
_V
6 elecg 6330 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
75, 6mpan 415 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
87exbidv 1753 . . 3  |-  ( A  e.  _V  ->  ( E. x  x  e.  [ A ] R  <->  E. x  A R x ) )
94, 8bitr4d 189 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R ) )
101, 3, 9pm5.21nii 655 1  |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1426    e. wcel 1438   _Vcvv 2619   class class class wbr 3845   dom cdm 4438   [cec 6290
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 3957  ax-pow 4009  ax-pr 4036
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 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-ec 6294
This theorem is referenced by:  ereldm  6335  elqsn0m  6360  ecelqsdm  6362
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