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Theorem ecdmn0m 6478
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 2700 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 ecexr 6441 . . 3  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
32exlimiv 1578 . 2  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
4 eldmg 4741 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
5 vex 2692 . . . . 5  |-  x  e. 
_V
6 elecg 6474 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
75, 6mpan 421 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
87exbidv 1798 . . 3  |-  ( A  e.  _V  ->  ( E. x  x  e.  [ A ] R  <->  E. x  A R x ) )
94, 8bitr4d 190 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R ) )
101, 3, 9pm5.21nii 694 1  |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1469    e. wcel 1481   _Vcvv 2689   class class class wbr 3936   dom cdm 4546   [cec 6434
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 4053  ax-pow 4105  ax-pr 4138
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 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-cnv 4554  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-ec 6438
This theorem is referenced by:  ereldm  6479  elqsn0m  6504  ecelqsdm  6506
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