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Theorem ecdmn0 6910
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 2928 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 n0 3601 . . 3  |-  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
)
3 ecexr 6873 . . . 4  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
43exlimiv 1641 . . 3  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
52, 4sylbi 188 . 2  |-  ( [ A ] R  =/=  (/)  ->  A  e.  _V )
6 vex 2923 . . . . 5  |-  x  e. 
_V
7 elecg 6906 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
86, 7mpan 652 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
98exbidv 1633 . . 3  |-  ( A  e.  _V  ->  ( E. x  x  e.  [ A ] R  <->  E. x  A R x ) )
102a1i 11 . . 3  |-  ( A  e.  _V  ->  ( [ A ] R  =/=  (/) 
<->  E. x  x  e. 
[ A ] R
) )
11 eldmg 5028 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
129, 10, 113bitr4rd 278 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) ) )
131, 5, 12pm5.21nii 343 1  |-  ( A  e.  dom  R  <->  [ A ] R  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1547    e. wcel 1721    =/= wne 2571   _Vcvv 2920   (/)c0 3592   class class class wbr 4176   dom cdm 4841   [cec 6866
This theorem is referenced by:  ereldm  6911  elqsn0  6936  ecelqsdm  6937  eceqoveq  6972  divsfval  13731  sylow1lem5  15195  vitalilem2  19458  vitalilem3  19459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-xp 4847  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-ec 6870
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