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Theorem relcoi2 4878
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 4623 . . . 4  |-  ( dom 
R  u.  ran  R
)  C_  U. U. R
2 unss 3147 . . . . 5  |-  ( ( dom  R  C_  U. U. R  /\  ran  R  C_  U.
U. R )  <->  ( dom  R  u.  ran  R ) 
C_  U. U. R )
3 simpr 108 . . . . 5  |-  ( ( dom  R  C_  U. U. R  /\  ran  R  C_  U.
U. R )  ->  ran  R  C_  U. U. R
)
42, 3sylbir 133 . . . 4  |-  ( ( dom  R  u.  ran  R )  C_  U. U. R  ->  ran  R  C_  U. U. R )
51, 4ax-mp 7 . . 3  |-  ran  R  C_ 
U. U. R
6 cores 4854 . . 3  |-  ( ran 
R  C_  U. U. R  ->  ( (  _I  |`  U. U. R )  o.  R
)  =  (  _I  o.  R ) )
75, 6mp1i 10 . 2  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  (  _I  o.  R ) )
8 coi2 4867 . 2  |-  ( Rel 
R  ->  (  _I  o.  R )  =  R )
97, 8eqtrd 2114 1  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    u. cun 2972    C_ wss 2974   U.cuni 3609    _I cid 4051   dom cdm 4371   ran crn 4372    |` cres 4373    o. ccom 4375   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383
This theorem is referenced by: (None)
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