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Theorem elrn 4919
Description: Membership in a range. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
elrn.1  |-  A  e. 
_V
Assertion
Ref Expression
elrn  |-  ( A  e.  ran  B  <->  E. x  x B A )
Distinct variable groups:    x, A    x, B

Proof of Theorem elrn
StepHypRef Expression
1 elrn.1 . . 3  |-  A  e. 
_V
21elrn2 4918 . 2  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
3 df-br 4024 . . 3  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43exbii 1569 . 2  |-  ( E. x  x B A  <->  E. x <. x ,  A >.  e.  B )
52, 4bitr4i 243 1  |-  ( A  e.  ran  B  <->  E. x  x B A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   ran crn 4690
This theorem is referenced by:  dmcosseq  4946  rnco  5179  dffo4  5676  fvclss  5760  rntpos  6247  fpwwe2lem11  8262  fpwwe2lem12  8263  fclim  12027  perfdvf  19253  dftr6  24107  dffr5  24110  brsset  24429  dfon3  24432  brtxpsd  24434  dffix2  24445  elsingles  24457  dfrdg4  24488  rnhmph  25534  inisegn0  27140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700
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