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Theorem elrn2g 4697
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 3674 . . . 4  |-  ( y  =  A  ->  <. x ,  y >.  =  <. x ,  A >. )
21eleq1d 2184 . . 3  |-  ( y  =  A  ->  ( <. x ,  y >.  e.  B  <->  <. x ,  A >.  e.  B ) )
32exbidv 1779 . 2  |-  ( y  =  A  ->  ( E. x <. x ,  y
>.  e.  B  <->  E. x <. x ,  A >.  e.  B ) )
4 dfrn3 4696 . 2  |-  ran  B  =  { y  |  E. x <. x ,  y
>.  e.  B }
53, 4elab2g 2802 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 104    = wceq 1314   E.wex 1451    e. wcel 1463   <.cop 3498   ran crn 4508
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  elrng  4698  fvelrn  5517  fo2ndf  6090
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