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Theorem elrn2 5111
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 3987 . . . 4  |-  ( y  =  A  ->  <. x ,  y >.  =  <. x ,  A >. )
32eleq1d 2504 . . 3  |-  ( y  =  A  ->  ( <. x ,  y >.  e.  B  <->  <. x ,  A >.  e.  B ) )
43exbidv 1637 . 2  |-  ( y  =  A  ->  ( E. x <. x ,  y
>.  e.  B  <->  E. x <. x ,  A >.  e.  B ) )
5 dfrn3 5062 . 2  |-  ran  B  =  { y  |  E. x <. x ,  y
>.  e.  B }
61, 4, 5elab2 3087 1  |-  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   ran crn 4881
This theorem is referenced by:  elrn  5112  dmrnssfld  5131  rniun  5284  ssrnres  5311  relssdmrn  5392  fvelrn  5868  tz7.48-1  6702  dfrn5  25403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-cnv 4888  df-dm 4890  df-rn 4891
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