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Theorem elrn 4777
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 4776 . 2  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
3 df-br 3925 . . 3  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
43exbii 1584 . 2  |-  ( E. x  x B A  <->  E. x <. x ,  A >.  e.  B )
52, 4bitr4i 186 1  |-  ( A  e.  ran  B  <->  E. x  x B A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1468    e. wcel 1480   _Vcvv 2681   <.cop 3525   class class class wbr 3924   ran crn 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-cnv 4542  df-dm 4544  df-rn 4545
This theorem is referenced by:  dmcosseq  4805  rnco  5040  dffo4  5561  rntpos  6147  fclim  11056  dvfgg  12811
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