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Theorem relrded 25742
Description: The range of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
Assertion
Ref Expression
relrded  |-  Rel  ran  Ded

Proof of Theorem relrded
StepHypRef Expression
1 strded 25739 . . . 4  |-  Ded  C_  (
( _V  X.  _V )  X.  ( _V  X.  _V ) )
2 rnss 4907 . . . 4  |-  ( Ded  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )  ->  ran  Ded  C_  ran  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) ) )
31, 2ax-mp 8 . . 3  |-  ran  Ded  C_ 
ran  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )
4 rnxpid 5109 . . 3  |-  ran  (
( _V  X.  _V )  X.  ( _V  X.  _V ) )  =  ( _V  X.  _V )
53, 4sseqtri 3210 . 2  |-  ran  Ded  C_  ( _V  X.  _V )
6 df-rel 4696 . 2  |-  ( Rel 
ran  Ded  <->  ran  Ded  C_  ( _V 
X.  _V ) )
75, 6mpbir 200 1  |-  Rel  ran  Ded
Colors of variables: wff set class
Syntax hints:   _Vcvv 2788    C_ wss 3152    X. cxp 4687   ran crn 4690   Rel wrel 4694   Dedcded 25734
This theorem is referenced by:  dedalg  25743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-ded 25735
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