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Theorem elrn2 4519
Description: Membership in a range. (Contributed by NM, 10-Jul-1994.)
Hypothesis
Ref Expression
elrn.1  _V
Assertion
Ref Expression
elrn2  ran  <. ,  >.
Distinct variable groups:   ,   ,

Proof of Theorem elrn2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elrn.1 . 2  _V
2 opeq2 3541 . . . 4  <. ,  >.  <. ,  >.
32eleq1d 2103 . . 3  <. ,  >.  <. ,  >.
43exbidv 1703 . 2  <. , 
>.  <. ,  >.
5 dfrn3 4467 . 2  ran  {  |  <. , 
>.  }
61, 4, 5elab2 2684 1  ran  <. ,  >.
Colors of variables: wff set class
Syntax hints:   wb 98   wceq 1242  wex 1378   wcel 1390   _Vcvv 2551   <.cop 3370   ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  elrn  4520  dmrnssfld  4538  rniun  4677  rnxpid  4698  ssrnres  4706  relssdmrn  4784
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