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Theorem eldm2 5059
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 5057 . 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 177   E.wex 1550    e. wcel 1725   _Vcvv 2948   <.cop 3809   dom cdm 4869
This theorem is referenced by:  dmss  5060  opeldm  5064  dmin  5068  dmiun  5069  dmuni  5070  dm0  5074  reldm0  5078  dmrnssfld  5120  dmcoss  5126  dmcosseq  5128  dmres  5158  iss  5180  dmsnopg  5332  relssdmrn  5381  funssres  5484  fun11iun  5686  dmfco  5788  axdc3lem2  8320  gsum2d2  15536  cnlnssadj  23571  eldm3  25374  dfdm5  25387  wfrlem12  25522  frrlem11  25548  tfrqfree  25746
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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