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Theorem relelrn 4684
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
Assertion
Ref Expression
relelrn  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  ran  R )

Proof of Theorem relelrn
StepHypRef Expression
1 brrelex 4491 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4492 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 109 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 brelrng 4679 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  B  e.  ran  R )
51, 2, 3, 4syl3anc 1175 1  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1439   _Vcvv 2620   class class class wbr 3851   ran crn 4453   Rel wrel 4457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-dm 4462  df-rn 4463
This theorem is referenced by:  relelrnb  4686  relelrni  4688  relfvssunirn  5334
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