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Theorem elrn2 4918
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 3797 . . . 4  |-  ( y  =  A  ->  <. x ,  y >.  =  <. x ,  A >. )
32eleq1d 2349 . . 3  |-  ( y  =  A  ->  ( <. x ,  y >.  e.  B  <->  <. x ,  A >.  e.  B ) )
43exbidv 1612 . 2  |-  ( y  =  A  ->  ( E. x <. x ,  y
>.  e.  B  <->  E. x <. x ,  A >.  e.  B ) )
5 dfrn3 4869 . 2  |-  ran  B  =  { y  |  E. x <. x ,  y
>.  e.  B }
61, 4, 5elab2 2917 1  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   ran crn 4690
This theorem is referenced by:  elrn  4919  dmrnssfld  4938  rniun  5091  ssrnres  5116  relssdmrn  5193  fvelrn  5661  tz7.48-1  6455  dfrn5  24133  dmrngcmp  25751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700
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