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Theorem ecdmn0m 6439
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 2671 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 ecexr 6402 . . 3  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
32exlimiv 1562 . 2  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
4 eldmg 4704 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
5 vex 2663 . . . . 5  |-  x  e. 
_V
6 elecg 6435 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
75, 6mpan 420 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
87exbidv 1781 . . 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 678 1  |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1453    e. wcel 1465   _Vcvv 2660   class class class wbr 3899   dom cdm 4509   [cec 6395
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399
This theorem is referenced by:  ereldm  6440  elqsn0m  6465  ecelqsdm  6467
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