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Theorem eldm2 4995
Description: Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
Hypothesis
Ref Expression
eldm.1  |-  A  e. 
_V
Assertion
Ref Expression
eldm2  |-  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B )
Distinct variable groups:    y, A    y, B

Proof of Theorem eldm2
StepHypRef Expression
1 eldm.1 . 2  |-  A  e. 
_V
2 eldm2g 4993 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
31, 2ax-mp 5 1  |-  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   E.wex 1657    e. wcel 1872   _Vcvv 3022   <.cop 3947   dom cdm 4796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-dm 4806
This theorem is referenced by:  dmss  4996  opeldm  5000  dmin  5004  dmiun  5005  dmuni  5006  dm0  5010  reldm0  5014  dmrnssfld  5055  dmcoss  5056  dmcosseq  5058  dmres  5087  iss  5114  dmsnopg  5269  relssdmrn  5318  funssres  5584  dmfco  5899  fun11iun  6711  wfrlem12  7002  axdc3lem2  8832  gsum2d2  17549  cnlnssadj  27675  prsdm  28672  eldm3  30353  dfdm5  30369  frrlem11  30477
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