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Theorem eldm2 4561
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 4559 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
31, 2ax-mp 7 1  |-  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1422    e. wcel 1434   _Vcvv 2602   <.cop 3409   dom cdm 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381
This theorem is referenced by:  dmss  4562  opeldm  4566  dmin  4571  dmiun  4572  dmuni  4573  dm0  4577  reldm0  4581  dmrnssfld  4623  dmcoss  4629  dmcosseq  4631  dmres  4660  iss  4684  dmxpss  4783  dmsnopg  4822  relssdmrn  4871  funssres  4972  fun11iun  5178  tfrlemibxssdm  5976  tfr1onlembxssdm  5992  tfrcllembxssdm  6005
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