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Theorem isprsr 25224
Description: The predicate "is a preset". (Contributed by FL, 1-May-2011.)
Assertion
Ref Expression
isprsr  |-  ( R  e.  A  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )

Proof of Theorem isprsr
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 releq 4771 . . 3  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 coeq1 4841 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  r ) )
3 coeq2 4842 . . . . 5  |-  ( r  =  R  ->  ( R  o.  r )  =  ( R  o.  R ) )
42, 3eqtrd 2315 . . . 4  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
5 sseq12 3201 . . . 4  |-  ( ( ( r  o.  r
)  =  ( R  o.  R )  /\  r  =  R )  ->  ( ( r  o.  r )  C_  r  <->  ( R  o.  R ) 
C_  R ) )
64, 5mpancom 650 . . 3  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
7 unieq 3836 . . . . . 6  |-  ( r  =  R  ->  U. r  =  U. R )
87unieqd 3838 . . . . 5  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
98reseq2d 4955 . . . 4  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  U. U. R ) )
10 sseq12 3201 . . . 4  |-  ( ( (  _I  |`  U. U. r )  =  (  _I  |`  U. U. R
)  /\  r  =  R )  ->  (
(  _I  |`  U. U. r )  C_  r  <->  (  _I  |`  U. U. R
)  C_  R )
)
119, 10mpancom 650 . . 3  |-  ( r  =  R  ->  (
(  _I  |`  U. U. r )  C_  r  <->  (  _I  |`  U. U. R
)  C_  R )
)
121, 6, 113anbi123d 1252 . 2  |-  ( r  =  R  ->  (
( Rel  r  /\  ( r  o.  r
)  C_  r  /\  (  _I  |`  U. U. r )  C_  r
)  <->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )
13 df-prs 25223 . 2  |- PresetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  (  _I  |`  U. U. r )  C_  r
) }
1412, 13elab2g 2916 1  |-  ( R  e.  A  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827    _I cid 4304    |` cres 4691    o. ccom 4693   Rel wrel 4694  PresetRelcpresetrel 25215
This theorem is referenced by:  preorel  25225  preodom2  25226  preoref12  25228  preoran2  25230  preotr1  25234  altprs2  25236  int2pre  25237  sqpre  25238  dupre1  25243  dfdir2  25291
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-co 4698  df-res 4701  df-prs 25223
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