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Theorem elrn 5086
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 5085 . 2  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
3 df-br 4418 . . 3  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43exbii 1712 . 2  |-  ( E. x  x B A  <->  E. x <. x ,  A >.  e.  B )
52, 4bitr4i 255 1  |-  ( A  e.  ran  B  <->  E. x  x B A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   E.wex 1659    e. wcel 1867   _Vcvv 3078   <.cop 3999   class class class wbr 4417   ran crn 4846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-cnv 4853  df-dm 4855  df-rn 4856
This theorem is referenced by:  dmcosseq  5107  inisegn0  5211  rnco  5352  dffo4  6044  fvclss  6153  rntpos  6985  fpwwe2lem11  9054  fpwwe2lem12  9055  fclim  13584  perfdvf  22765  dftr6  30218  dffr5  30221  brsset  30482  dfon3  30485  brtxpsd  30487  dffix2  30498  elsingles  30511  dfrdg4  30544
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