MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldmg Unicode version

Theorem eldmg 4874
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 4026 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 1612 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4699 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 2916 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   class class class wbr 4023   dom cdm 4689
This theorem is referenced by:  eldm2g  4875  eldm  4876  breldmg  4884  releldmb  4913  funeu  5278  fneu  5348  ndmfv  5552  erref  6680  ecdmn0  6702  rlimdm  12025  rlimdmo1  12091  iscmet3lem2  18718  dvcnp2  19269  ulmcau  19772  pserulm  19798  mulog2sum  20686
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  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-dm 4699
  Copyright terms: Public domain W3C validator