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Theorem reflincror 25112
Description: If a relation  R is reflexive, it is included in  ( R  o.  R ). (Contributed by FL, 8-May-2011.)
Assertion
Ref Expression
reflincror  |-  ( ( Rel  R  /\  (  _I  |`  U. U. R
)  C_  R )  ->  R  C_  ( R  o.  R ) )

Proof of Theorem reflincror
StepHypRef Expression
1 relcoi2 5200 . . 3  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  R )
2 coss1 4839 . . 3  |-  ( (  _I  |`  U. U. R
)  C_  R  ->  ( (  _I  |`  U. U. R )  o.  R
)  C_  ( R  o.  R ) )
3 sseq1 3199 . . . 4  |-  ( ( (  _I  |`  U. U. R )  o.  R
)  =  R  -> 
( ( (  _I  |`  U. U. R )  o.  R )  C_  ( R  o.  R
)  <->  R  C_  ( R  o.  R ) ) )
43biimpd 198 . . 3  |-  ( ( (  _I  |`  U. U. R )  o.  R
)  =  R  -> 
( ( (  _I  |`  U. U. R )  o.  R )  C_  ( R  o.  R
)  ->  R  C_  ( R  o.  R )
) )
51, 2, 4syl2im 34 . 2  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  C_  R  ->  R 
C_  ( R  o.  R ) ) )
65imp 418 1  |-  ( ( Rel  R  /\  (  _I  |`  U. U. R
)  C_  R )  ->  R  C_  ( R  o.  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    C_ wss 3152   U.cuni 3827    _I cid 4304    |` cres 4691    o. ccom 4693   Rel wrel 4694
This theorem is referenced by:  altprs2  25236
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
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