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Theorem opelrng 4615
Description: Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)
Assertion
Ref Expression
opelrng  |-  ( ( A  e.  F  /\  B  e.  G  /\  <. A ,  B >.  e.  C )  ->  B  e.  ran  C )

Proof of Theorem opelrng
StepHypRef Expression
1 df-br 3807 . 2  |-  ( A C B  <->  <. A ,  B >.  e.  C )
2 brelrng 4614 . 2  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  ran  C )
31, 2syl3an3br 1211 1  |-  ( ( A  e.  F  /\  B  e.  G  /\  <. A ,  B >.  e.  C )  ->  B  e.  ran  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 920    e. wcel 1434   <.cop 3420   class class class wbr 3806   ran crn 4393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-cnv 4400  df-dm 4402  df-rn 4403
This theorem is referenced by:  2ndrn  5861
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