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Theorem opelrn 4910
Description: Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
Hypotheses
Ref Expression
brelrn.1  |-  A  e. 
_V
brelrn.2  |-  B  e. 
_V
Assertion
Ref Expression
opelrn  |-  ( <. A ,  B >.  e.  C  ->  B  e.  ran  C )

Proof of Theorem opelrn
StepHypRef Expression
1 df-br 4024 . 2  |-  ( A C B  <->  <. A ,  B >.  e.  C )
2 brelrn.1 . . 3  |-  A  e. 
_V
3 brelrn.2 . . 3  |-  B  e. 
_V
42, 3brelrn 4909 . 2  |-  ( A C B  ->  B  e.  ran  C )
51, 4sylbir 204 1  |-  ( <. A ,  B >.  e.  C  ->  B  e.  ran  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   ran crn 4690
This theorem is referenced by:  zfrep6  5748  2ndrn  6168  disjen  7018  r0weon  7640  gsum2d  15223  dfres3  24116
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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