ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecdmn0m Unicode version

Theorem ecdmn0m 6567
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 2746 . 2  |-  ( A  e.  dom  R  ->  A  e.  _V )
2 ecexr 6530 . . 3  |-  ( x  e.  [ A ] R  ->  A  e.  _V )
32exlimiv 1596 . 2  |-  ( E. x  x  e.  [ A ] R  ->  A  e.  _V )
4 eldmg 4815 . . 3  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
5 vex 2738 . . . . 5  |-  x  e. 
_V
6 elecg 6563 . . . . 5  |-  ( ( x  e.  _V  /\  A  e.  _V )  ->  ( x  e.  [ A ] R  <->  A R x ) )
75, 6mpan 424 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  [ A ] R  <->  A R x ) )
87exbidv 1823 . . 3  |-  ( A  e.  _V  ->  ( E. x  x  e.  [ A ] R  <->  E. x  A R x ) )
94, 8bitr4d 191 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R ) )
101, 3, 9pm5.21nii 704 1  |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   E.wex 1490    e. wcel 2146   _Vcvv 2735   class class class wbr 3998   dom cdm 4620   [cec 6523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-ec 6527
This theorem is referenced by:  ereldm  6568  elqsn0m  6593  ecelqsdm  6595
  Copyright terms: Public domain W3C validator