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Theorem dupre1 25243
Description: The converse of a preset is a preset. The case  ( `' R  e. PresetRel  ->  R  e. PresetRel ) is true only if 
R is a relation. See dupre2 25244. (Contributed by FL, 5-Jan-2009.)
Assertion
Ref Expression
dupre1  |-  ( R  e. PresetRel  ->  `' R  e. PresetRel )

Proof of Theorem dupre1
StepHypRef Expression
1 relcnv 5051 . . . . . 6  |-  Rel  `' R
21a1ii 24 . . . . 5  |-  ( R  e. PresetRel  ->  ( Rel  R  ->  Rel  `' R ) )
3 preorel 25225 . . . . . 6  |-  ( R  e. PresetRel  ->  Rel  R )
43a1d 22 . . . . 5  |-  ( R  e. PresetRel  ->  ( Rel  `' R  ->  Rel  R )
)
52, 4impbid 183 . . . 4  |-  ( R  e. PresetRel  ->  ( Rel  R  <->  Rel  `' R ) )
6 relcnvtr 5192 . . . . 5  |-  ( Rel 
R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
73, 6syl 15 . . . 4  |-  ( R  e. PresetRel  ->  ( ( R  o.  R )  C_  R 
<->  ( `' R  o.  `' R )  C_  `' R ) )
8 relrefcnv 25117 . . . . 5  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  C_  R  <->  (  _I  |` 
U. U. `' R ) 
C_  `' R ) )
93, 8syl 15 . . . 4  |-  ( R  e. PresetRel  ->  ( (  _I  |`  U. U. R ) 
C_  R  <->  (  _I  |` 
U. U. `' R ) 
C_  `' R ) )
105, 7, 93anbi123d 1252 . . 3  |-  ( R  e. PresetRel  ->  ( ( 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 ) ) )
11 isprsr 25224 . . 3  |-  ( R  e. PresetRel  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )
12 cnvexg 5208 . . . 4  |-  ( R  e. PresetRel  ->  `' R  e. 
_V )
13 isprsr 25224 . . . 4  |-  ( `' R  e.  _V  ->  ( `' R  e. PresetRel  <->  ( Rel  `' R  /\  ( `' R  o.  `' R
)  C_  `' R  /\  (  _I  |`  U. U. `' R )  C_  `' R ) ) )
1412, 13syl 15 . . 3  |-  ( R  e. PresetRel  ->  ( `' R  e. PresetRel  <->  ( Rel  `' R  /\  ( `' R  o.  `' R )  C_  `' R  /\  (  _I  |`  U. U. `' R )  C_  `' R ) ) )
1510, 11, 143bitr4d 276 . 2  |-  ( R  e. PresetRel  ->  ( R  e. PresetRel  <->  `' R  e. PresetRel ) )
1615ibi 232 1  |-  ( R  e. PresetRel  ->  `' R  e. PresetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1684   _Vcvv 2788    C_ wss 3152   U.cuni 3827    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   Rel wrel 4694  PresetRelcpresetrel 25215
This theorem is referenced by:  dupre2  25244  mnlmxl2  25269  mxlmnl2  25270
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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  df-prs 25223
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