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Theorem relrcat 25866
Description: The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)
Assertion
Ref Expression
relrcat  |-  Rel  ran  Cat
OLD

Proof of Theorem relrcat
StepHypRef Expression
1 strcat 25863 . . . 4  |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
2 rnss 4923 . . . 4  |-  (  Cat
OLD  C_  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )  ->  ran  Cat OLD  C_  ran  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) ) )
31, 2ax-mp 8 . . 3  |-  ran  Cat OLD  C_  ran  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )
4 rnxpid 5125 . . 3  |-  ran  (
( _V  X.  _V )  X.  ( _V  X.  _V ) )  =  ( _V  X.  _V )
53, 4sseqtri 3223 . 2  |-  ran  Cat OLD  C_  ( _V  X.  _V )
6 df-rel 4712 . 2  |-  ( Rel 
ran  Cat OLD  <->  ran  Cat OLD  C_  ( _V  X.  _V ) )
75, 6mpbir 200 1  |-  Rel  ran  Cat
OLD
Colors of variables: wff set class
Syntax hints:   _Vcvv 2801    C_ wss 3165    X. cxp 4703   ran crn 4706   Rel wrel 4710    Cat OLD ccatOLD 25855
This theorem is referenced by:  catded  25867
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-ral 2561  df-rex 2562  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-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-catOLD 25856
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