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Theorem relelrn 5103
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 4916 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4917 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 448 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 brelrng 5099 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  B  e.  ran  R )
51, 2, 3, 4syl3anc 1184 1  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   _Vcvv 2956   class class class wbr 4212   ran crn 4879   Rel wrel 4883
This theorem is referenced by:  relelrnb  5105  relelrni  5107  spwpr4  14663  spwpr4c  14664  dirge  14682  metideq  24288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889
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