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Theorem opelrn 4926
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 4040 . 2  |-  ( A C B  <->  <. A ,  B >.  e.  C )
2 brelrn.1 . . 3  |-  A  e. 
_V
3 brelrn.2 . . 3  |-  B  e. 
_V
42, 3brelrn 4925 . 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 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   ran crn 4706
This theorem is referenced by:  zfrep6  5764  2ndrn  6184  disjen  7034  r0weon  7656  gsum2d  15239  dfres3  24187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716
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