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Theorem brelrn 4925
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 4924 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A C B )  ->  B  e.  ran  C )
41, 2, 3mp3an12 1267 1  |-  ( A C B  ->  B  e.  ran  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   ran crn 4706
This theorem is referenced by:  opelrn  4926  dfco2a  5189  cores  5192  dffun9  5298  funcnv  5326  rntpos  6263  aceq3lem  7763  axdclem  8162  axdclem2  8163  shftfval  11581  psdmrn  14332  itg1addlem4  19070  ranfldrefc  25161  domrngref  25163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716
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