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Theorem eldm 5058
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
eldm.1  |-  A  e. 
_V
Assertion
Ref Expression
eldm  |-  ( A  e.  dom  B  <->  E. y  A B y )
Distinct variable groups:    y, A    y, B

Proof of Theorem eldm
StepHypRef Expression
1 eldm.1 . 2  |-  A  e. 
_V
2 eldmg 5056 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
31, 2ax-mp 8 1  |-  ( A  e.  dom  B  <->  E. y  A B y )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550    e. wcel 1725   _Vcvv 2948   class class class wbr 4204   dom cdm 4869
This theorem is referenced by:  dmi  5075  dmcoss  5126  dmcosseq  5128  dminss  5277  dmsnn0  5326  dffun7  5470  dffun8  5471  fnres  5552  fndmdif  5825  dff3  5873  frxp  6447  reldmtpos  6478  dmtpos  6482  opabiota  6529  aceq3lem  7990  axdc2lem  8317  axdclem2  8389  fpwwe2lem12  8505  nqerf  8796  shftdm  11874  xpsfrnel2  13778  bcthlem4  19268  dchrisumlem3  21173  eupath  21691  fundmpss  25377  elfix  25698  fnsingle  25714  fnimage  25724  funpartlem  25737  dfrdg4  25745  prtlem16  26655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-dm 4879
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