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Theorem elrn2 4781
Description: Membership in a range. (Contributed by NM, 10-Jul-1994.)
Hypothesis
Ref Expression
elrn.1  |-  A  e. 
_V
Assertion
Ref Expression
elrn2  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem elrn2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elrn.1 . 2  |-  A  e. 
_V
2 opeq2 3706 . . . 4  |-  ( y  =  A  ->  <. x ,  y >.  =  <. x ,  A >. )
32eleq1d 2208 . . 3  |-  ( y  =  A  ->  ( <. x ,  y >.  e.  B  <->  <. x ,  A >.  e.  B ) )
43exbidv 1797 . 2  |-  ( y  =  A  ->  ( E. x <. x ,  y
>.  e.  B  <->  E. x <. x ,  A >.  e.  B ) )
5 dfrn3 4728 . 2  |-  ran  B  =  { y  |  E. x <. x ,  y
>.  e.  B }
61, 4, 5elab2 2832 1  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2686   <.cop 3530   ran crn 4540
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550
This theorem is referenced by:  elrn  4782  dmrnssfld  4802  rniun  4949  rnxpid  4973  ssrnres  4981  relssdmrn  5059
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