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Theorem opelrn 5240
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 4454 . 2  |-  ( A C B  <->  <. A ,  B >.  e.  C )
2 brelrn.1 . . 3  |-  A  e. 
_V
3 brelrn.2 . . 3  |-  B  e. 
_V
42, 3brelrn 5239 . 2  |-  ( A C B  ->  B  e.  ran  C )
51, 4sylbir 213 1  |-  ( <. A ,  B >.  e.  C  ->  B  e.  ran  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   _Vcvv 3118   <.cop 4039   class class class wbr 4453   ran crn 5006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-cnv 5013  df-dm 5015  df-rn 5016
This theorem is referenced by:  zfrep6  6763  2ndrn  6843  disjen  7686  r0weon  8402  gsum2dlem1  16868  gsum2dlem2  16869  gsum2dOLD  16871  dfres3  29122
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