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Theorem eldm2g 4875
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 4874 . 2  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
2 df-br 4024 . . 3  |-  ( A B y  <->  <. A , 
y >.  e.  B )
32exbii 1569 . 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 1528    e. wcel 1684   <.cop 3643   class class class wbr 4023   dom cdm 4689
This theorem is referenced by:  eldm2  4877  dmfco  5593  releldm2  6170  tfrlem9  6401  climcau  12144  caucvgb  12152  lmff  17029  axhcompl-zf  21578
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
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