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Theorem opelrn 5086
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 4308 . 2  |-  ( A C B  <->  <. A ,  B >.  e.  C )
2 brelrn.1 . . 3  |-  A  e. 
_V
3 brelrn.2 . . 3  |-  B  e. 
_V
42, 3brelrn 5085 . 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 1756   _Vcvv 2987   <.cop 3898   class class class wbr 4307   ran crn 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-rab 2739  df-v 2989  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-br 4308  df-opab 4366  df-cnv 4863  df-dm 4865  df-rn 4866
This theorem is referenced by:  zfrep6  6560  2ndrn  6637  disjen  7483  r0weon  8194  gsum2dlem1  16476  gsum2dlem2  16477  gsum2dOLD  16479  dfres3  27584
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