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Theorem relelrn 4912
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 4727 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4728 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 447 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 brelrng 4908 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  B  e.  ran  R )
51, 2, 3, 4syl3anc 1182 1  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   ran crn 4690   Rel wrel 4694
This theorem is referenced by:  relelrnb  4914  relelrni  4916  spwpr4  14340  spwpr4c  14341  dirge  14359  pre2befi2  25232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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