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Theorem eldm2g 4891
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 4890 . 2  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
2 df-br 4040 . . 3  |-  ( A B y  <->  <. A , 
y >.  e.  B )
32exbii 1572 . 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 1531    e. wcel 1696   <.cop 3656   class class class wbr 4039   dom cdm 4705
This theorem is referenced by:  eldm2  4893  dmfco  5609  releldm2  6186  tfrlem9  6417  climcau  12160  caucvgb  12168  lmff  17045  axhcompl-zf  21594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-dm 4715
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