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Theorem eldm2g 4957
Description: Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
eldm2g  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
Distinct variable groups:    y, A    y, B
Allowed substitution hint:    V( y)

Proof of Theorem eldm2g
StepHypRef Expression
1 eldmg 4956 . 2  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
2 df-br 4105 . . 3  |-  ( A B y  <->  <. A , 
y >.  e.  B )
32exbii 1582 . 2  |-  ( E. y  A B y  <->  E. y <. A ,  y
>.  e.  B )
41, 3syl6bb 252 1  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1541    e. wcel 1710   <.cop 3719   class class class wbr 4104   dom cdm 4771
This theorem is referenced by:  eldm2  4959  dmfco  5676  releldm2  6257  tfrlem9  6488  climcau  12240  caucvgb  12249  lmff  17135  axhcompl-zf  21692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-dm 4781
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