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Theorem elrn2 5235
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 4209 . . . 4  |-  ( y  =  A  ->  <. x ,  y >.  =  <. x ,  A >. )
32eleq1d 2531 . . 3  |-  ( y  =  A  ->  ( <. x ,  y >.  e.  B  <->  <. x ,  A >.  e.  B ) )
43exbidv 1685 . 2  |-  ( y  =  A  ->  ( E. x <. x ,  y
>.  e.  B  <->  E. x <. x ,  A >.  e.  B ) )
5 dfrn3 5185 . 2  |-  ran  B  =  { y  |  E. x <. x ,  y
>.  e.  B }
61, 4, 5elab2 3248 1  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374   E.wex 1591    e. wcel 1762   _Vcvv 3108   <.cop 4028   ran crn 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-cnv 5002  df-dm 5004  df-rn 5005
This theorem is referenced by:  elrn  5236  dmrnssfld  5254  rniun  5409  ssrnres  5438  relssdmrn  5521  fvelrn  6010  tz7.48-1  7100  prsrn  27521  dfrn5  28772
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