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Theorem reflincror 25215
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 5216 . . 3  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  o.  R )  =  R )
2 coss1 4855 . . 3  |-  ( (  _I  |`  U. U. R
)  C_  R  ->  ( (  _I  |`  U. U. R )  o.  R
)  C_  ( R  o.  R ) )
3 sseq1 3212 . . . 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 1632    C_ wss 3165   U.cuni 3843    _I cid 4320    |` cres 4707    o. ccom 4709   Rel wrel 4710
This theorem is referenced by:  altprs2  25339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717
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