MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relcoi2 Unicode version

Theorem relcoi2 5200
Description: Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
Assertion
Ref Expression
relcoi2  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  R )

Proof of Theorem relcoi2
StepHypRef Expression
1 dmrnssfld 4938 . . . 4  |-  ( dom 
R  u.  ran  R
)  C_  U. U. R
2 unss 3349 . . . . 5  |-  ( ( dom  R  C_  U. U. R  /\  ran  R  C_  U.
U. R )  <->  ( dom  R  u.  ran  R ) 
C_  U. U. R )
3 simpr 447 . . . . 5  |-  ( ( dom  R  C_  U. U. R  /\  ran  R  C_  U.
U. R )  ->  ran  R  C_  U. U. R
)
42, 3sylbir 204 . . . 4  |-  ( ( dom  R  u.  ran  R )  C_  U. U. R  ->  ran  R  C_  U. U. R )
51, 4ax-mp 8 . . 3  |-  ran  R  C_ 
U. U. R
6 cores 5176 . . 3  |-  ( ran 
R  C_  U. U. R  ->  ( (  _I  |`  U. U. R )  o.  R
)  =  (  _I  o.  R ) )
75, 6mp1i 11 . 2  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  (  _I  o.  R ) )
8 coi2 5189 . 2  |-  ( Rel 
R  ->  (  _I  o.  R )  =  R )
97, 8eqtrd 2315 1  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    u. cun 3150    C_ wss 3152   U.cuni 3827    _I cid 4304   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694
This theorem is referenced by:  tsrdir  14360  relexp1  24027  reflincror  25112
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-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701
  Copyright terms: Public domain W3C validator