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Theorem ecdmn0m 6235
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 2619 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 ecexr 6198 . . 3  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
32exlimiv 1530 . 2  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
4 eldmg 4578 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
5 vex 2613 . . . . 5  |-  x  e. 
_V
6 elecg 6231 . . . . 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 1748 . . 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 653 1  |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1422    e. wcel 1434   _Vcvv 2610   class class class wbr 3805   dom cdm 4391   [cec 6191
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 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-ec 6195
This theorem is referenced by:  ereldm  6236  elqsn0m  6261  ecelqsdm  6263
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