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Theorem relcoi2 5381
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 5111 . . . 4  |-  ( dom 
R  u.  ran  R
)  C_  U. U. R
2 unss 3619 . . . . 5  |-  ( ( dom  R  C_  U. U. R  /\  ran  R  C_  U.
U. R )  <->  ( dom  R  u.  ran  R ) 
C_  U. U. R )
3 simpr 467 . . . . 5  |-  ( ( dom  R  C_  U. U. R  /\  ran  R  C_  U.
U. R )  ->  ran  R  C_  U. U. R
)
42, 3sylbir 218 . . . 4  |-  ( ( dom  R  u.  ran  R )  C_  U. U. R  ->  ran  R  C_  U. U. R )
51, 4ax-mp 5 . . 3  |-  ran  R  C_ 
U. U. R
6 cores 5356 . . 3  |-  ( ran 
R  C_  U. U. R  ->  ( (  _I  |`  U. U. R )  o.  R
)  =  (  _I  o.  R ) )
75, 6mp1i 13 . 2  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  (  _I  o.  R ) )
8 coi2 5370 . 2  |-  ( Rel 
R  ->  (  _I  o.  R )  =  R )
97, 8eqtrd 2495 1  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    u. cun 3413    C_ wss 3415   U.cuni 4211    _I cid 4762   dom cdm 4852   ran crn 4853    |` cres 4854    o. ccom 4856   Rel wrel 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864
This theorem is referenced by:  relexpsucr  13140  tsrdir  16532
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