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Theorem brelrn 5033
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
Hypotheses
Ref Expression
brelrn.1  |-  A  e. 
_V
brelrn.2  |-  B  e. 
_V
Assertion
Ref Expression
brelrn  |-  ( A C B  ->  B  e.  ran  C )

Proof of Theorem brelrn
StepHypRef Expression
1 brelrn.1 . 2  |-  A  e. 
_V
2 brelrn.2 . 2  |-  B  e. 
_V
3 brelrng 5032 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A C B )  ->  B  e.  ran  C )
41, 2, 3mp3an12 1269 1  |-  ( A C B  ->  B  e.  ran  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   _Vcvv 2892   class class class wbr 4146   ran crn 4812
This theorem is referenced by:  opelrn  5034  dfco2a  5303  cores  5306  dffun9  5414  funcnv  5444  rntpos  6421  aceq3lem  7927  axdclem  8325  axdclem2  8326  shftfval  11805  psdmrn  14559  metustexhalf  18469  itg1addlem4  19451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-cnv 4819  df-dm 4821  df-rn 4822
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