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Theorem elrng 5062
Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
elrng  |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x  x B A ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem elrng
StepHypRef Expression
1 elrn2g 5061 . 2  |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B ) )
2 df-br 4213 . . 3  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
32exbii 1592 . 2  |-  ( E. x  x B A  <->  E. x <. x ,  A >.  e.  B )
41, 3syl6bbr 255 1  |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x  x B A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1550    e. wcel 1725   <.cop 3817   class class class wbr 4212   ran crn 4879
This theorem is referenced by:  relelrnb  5105  trpredpred  25506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889
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