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Theorem elrng 4554
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 4553 . 2  |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B ) )
2 df-br 3794 . . 3  |-  ( x B A  <->  <. x ,  A >.  e.  B
)
32exbii 1537 . 2  |-  ( E. x  x B A  <->  E. x <. x ,  A >.  e.  B )
41, 3syl6bbr 196 1  |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x  x B A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   E.wex 1422    e. wcel 1434   <.cop 3409   class class class wbr 3793   ran crn 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by:  relelrnb  4600
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