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Theorem relelrn 4775
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 4579 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4580 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 109 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 brelrng 4770 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  B  e.  ran  R )
51, 2, 3, 4syl3anc 1216 1  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480   _Vcvv 2686   class class class wbr 3929   ran crn 4540   Rel wrel 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550
This theorem is referenced by:  relelrnb  4777  relelrni  4779  relfvssunirn  5437
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