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

Theorem eldmg 4558
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
eldmg  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Distinct variable groups:    y, A    y, B
Allowed substitution hint:    V( y)

Proof of Theorem eldmg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 breq1 3796 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1747 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4381 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 2741 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   class class class wbr 3793   dom cdm 4371
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381
This theorem is referenced by:  eldm2g  4559  eldm  4560  breldmg  4569  releldmb  4599  funeu  4956  fneu  5034  ndmfvg  5236  erref  6192  ecdmn0m  6214  shftdm  9848
  Copyright terms: Public domain W3C validator