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Theorem elrn2g 4626
Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
elrn2g  |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem elrn2g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 opeq2 3623 . . . 4  |-  ( y  =  A  ->  <. x ,  y >.  =  <. x ,  A >. )
21eleq1d 2156 . . 3  |-  ( y  =  A  ->  ( <. x ,  y >.  e.  B  <->  <. x ,  A >.  e.  B ) )
32exbidv 1753 . 2  |-  ( y  =  A  ->  ( E. x <. x ,  y
>.  e.  B  <->  E. x <. x ,  A >.  e.  B ) )
4 dfrn3 4625 . 2  |-  ran  B  =  { y  |  E. x <. x ,  y
>.  e.  B }
53, 4elab2g 2762 1  |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   <.cop 3449   ran crn 4439
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  elrng  4627  fvelrn  5430  fo2ndf  5992
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