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Theorem brelrn 5092
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 5091 . 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 1725   _Vcvv 2948   class class class wbr 4204   ran crn 4871
This theorem is referenced by:  opelrn  5093  dfco2a  5362  cores  5365  dffun9  5473  funcnv  5503  rntpos  6484  aceq3lem  7993  axdclem  8391  axdclem2  8392  shftfval  11877  psdmrn  14631  metustexhalfOLD  18585  metustexhalf  18586  itg1addlem4  19583
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cnv 4878  df-dm 4880  df-rn 4881
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