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Theorem eldm2 4893
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 4891 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
31, 2ax-mp 8 1  |-  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1531    e. wcel 1696   _Vcvv 2801   <.cop 3656   dom cdm 4705
This theorem is referenced by:  dmss  4894  opeldm  4898  dmin  4902  dmiun  4903  dmuni  4904  dm0  4908  reldm0  4912  dmrnssfld  4954  dmcoss  4960  dmcosseq  4962  dmres  4992  iss  5014  dmsnopg  5160  relssdmrn  5209  funssres  5310  fun11iun  5509  dmfco  5609  axdc3lem2  8093  gsum2d2  15241  cnlnssadj  22676  eldm3  24190  dfdm5  24203  wfrlem12  24338  frrlem11  24364  tfrqfree  24561  dmrngcmp  25854
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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