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Theorem elrn 5243
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 5242 . 2  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
3 df-br 4448 . . 3  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43exbii 1644 . 2  |-  ( E. x  x B A  <->  E. x <. x ,  A >.  e.  B )
52, 4bitr4i 252 1  |-  ( A  e.  ran  B  <->  E. x  x B A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1596    e. wcel 1767   _Vcvv 3113   <.cop 4033   class class class wbr 4447   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  dmcosseq  5264  rnco  5513  dffo4  6038  fvclss  6143  rntpos  6969  fpwwe2lem11  9019  fpwwe2lem12  9020  fclim  13342  perfdvf  22134  dftr6  29032  dffr5  29035  brsset  29392  dfon3  29395  brtxpsd  29397  dffix2  29408  elsingles  29421  dfrdg4  29453  inisegn0  30820
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