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Theorem ecdmn0 6702
Description: A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecdmn0  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )

Proof of Theorem ecdmn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 n0 3464 . . 3  |-  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
)
3 ecexr 6665 . . . 4  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
43exlimiv 1666 . . 3  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
52, 4sylbi 187 . 2  |-  ( [ A ] R  =/=  (/)  ->  A  e.  _V )
6 vex 2791 . . . . 5  |-  x  e. 
_V
7 elecg 6698 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
86, 7mpan 651 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
98exbidv 1612 . . 3  |-  ( A  e.  _V  ->  ( E. x  x  e.  [ A ] R  <->  E. x  A R x ) )
102a1i 10 . . 3  |-  ( A  e.  _V  ->  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
) )
11 eldmg 4874 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
129, 10, 113bitr4rd 277 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) ) )
131, 5, 12pm5.21nii 342 1  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   class class class wbr 4023   dom cdm 4689   [cec 6658
This theorem is referenced by:  ereldm  6703  elqsn0  6728  ecelqsdm  6729  eceqoveq  6763  divsfval  13449  sylow1lem5  14913  vitalilem2  18964  vitalilem3  18965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662
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