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Theorem eldm2g 5025
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 5024 . 2  |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
2 df-br 4173 . . 3  |-  ( A B y  <->  <. A , 
y >.  e.  B )
32exbii 1589 . 2  |-  ( E. y  A B y  <->  E. y <. A ,  y
>.  e.  B )
41, 3syl6bb 253 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 177   E.wex 1547    e. wcel 1721   <.cop 3777   class class class wbr 4172   dom cdm 4837
This theorem is referenced by:  eldm2  5027  dmfco  5756  releldm2  6356  tfrlem9  6605  climcau  12419  caucvgb  12428  lmff  17319  axhcompl-zf  22454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-dm 4847
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