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Theorem eldm2g 4703
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 4702 . 2  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
2 df-br 3898 . . 3  |-  ( A B y  <->  <. A , 
y >.  e.  B )
32exbii 1567 . 2  |-  ( E. y  A B y  <->  E. y <. A ,  y
>.  e.  B )
41, 3syl6bb 195 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 104   E.wex 1451    e. wcel 1463   <.cop 3498   class class class wbr 3897   dom cdm 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-dm 4517
This theorem is referenced by:  eldm2  4705  opeldmg  4712  dmfco  5455  releldm2  6049  tfrlem9  6182  climcau  11056  lmff  12313
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