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

Theorem eldmg 4781
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
StepHypRef Expression
1 breq1 3923 . . 3  |-  ( x  =  A  ->  (
x B y  <->  A B
y ) )
21exbidv 2005 . 2  |-  ( x  =  A  ->  ( E. y  x B
y  <->  E. y  A B y ) )
3 df-dm 4598 . 2  |-  dom  B  =  { x  |  E. y  x B y }
42, 3elab2g 2853 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   E.wex 1537    = wceq 1619    e. wcel 1621   class class class wbr 3920   dom cdm 4580
This theorem is referenced by:  eldm2g  4782  eldm  4783  breldmg  4791  releldmb  4820  funeu  5136  fneu  5205  ndmfv  5405  erref  6566  ecdmn0  6588  rlimdm  11902  rlimdmo1  11968  iscmet3lem2  18550  dvcnp2  19101  ulmcau  19604  pserulm  19630  mulog2sum  20518
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-dm 4598
  Copyright terms: Public domain W3C validator