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Theorem relcoi2 5397
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 5129 . . . 4  |-  ( dom 
R  u.  ran  R
)  C_  U. U. R
2 unss 3521 . . . . 5  |-  ( ( dom  R  C_  U. U. R  /\  ran  R  C_  U.
U. R )  <->  ( dom  R  u.  ran  R ) 
C_  U. U. R )
3 simpr 448 . . . . 5  |-  ( ( dom  R  C_  U. U. R  /\  ran  R  C_  U.
U. R )  ->  ran  R  C_  U. U. R
)
42, 3sylbir 205 . . . 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 5373 . . 3  |-  ( ran 
R  C_  U. U. R  ->  ( (  _I  |`  U. U. R )  o.  R
)  =  (  _I  o.  R ) )
75, 6mp1i 12 . 2  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  (  _I  o.  R ) )
8 coi2 5386 . 2  |-  ( Rel 
R  ->  (  _I  o.  R )  =  R )
97, 8eqtrd 2468 1  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    u. cun 3318    C_ wss 3320   U.cuni 4015    _I cid 4493   dom cdm 4878   ran crn 4879    |` cres 4880    o. ccom 4882   Rel wrel 4883
This theorem is referenced by:  tsrdir  14683  relexp1  25131
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890
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