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Theorem elrn2 4598
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 3573 . . . 4  |-  ( y  =  A  ->  <. x ,  y >.  =  <. x ,  A >. )
32eleq1d 2148 . . 3  |-  ( y  =  A  ->  ( <. x ,  y >.  e.  B  <->  <. x ,  A >.  e.  B ) )
43exbidv 1747 . 2  |-  ( y  =  A  ->  ( E. x <. x ,  y
>.  e.  B  <->  E. x <. x ,  A >.  e.  B ) )
5 dfrn3 4546 . 2  |-  ran  B  =  { y  |  E. x <. x ,  y
>.  e.  B }
61, 4, 5elab2 2742 1  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602   <.cop 3403   ran crn 4366
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 3898  ax-pow 3950  ax-pr 3966
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 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-cnv 4373  df-dm 4375  df-rn 4376
This theorem is referenced by:  elrn  4599  dmrnssfld  4617  rniun  4758  rnxpid  4779  ssrnres  4787  relssdmrn  4865
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