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Theorem dispos 25287
Description: A restriction of the identity is a poset. (Contributed by FL, 2-Aug-2009.)
Assertion
Ref Expression
dispos  |-  ( A  e.  V  ->  (  _I  |`  A )  e.  PosetRel )

Proof of Theorem dispos
StepHypRef Expression
1 relres 4983 . . 3  |-  Rel  (  _I  |`  A )
2 f1oi 5511 . . . . . 6  |-  (  _I  |`  A ) : A -1-1-onto-> A
3 f1of 5472 . . . . . 6  |-  ( (  _I  |`  A ) : A -1-1-onto-> A  ->  (  _I  |`  A ) : A --> A )
42, 3ax-mp 8 . . . . 5  |-  (  _I  |`  A ) : A --> A
5 fcoi1 5415 . . . . 5  |-  ( (  _I  |`  A ) : A --> A  ->  (
(  _I  |`  A )  o.  (  _I  |`  A ) )  =  (  _I  |`  A ) )
64, 5ax-mp 8 . . . 4  |-  ( (  _I  |`  A )  o.  (  _I  |`  A ) )  =  (  _I  |`  A )
76eqimssi 3232 . . 3  |-  ( (  _I  |`  A )  o.  (  _I  |`  A ) )  C_  (  _I  |`  A )
8 cnvresid 5322 . . . . 5  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
98ineq2i 3367 . . . 4  |-  ( (  _I  |`  A )  i^i  `' (  _I  |`  A ) )  =  ( (  _I  |`  A )  i^i  (  _I  |`  A ) )
10 inidm 3378 . . . 4  |-  ( (  _I  |`  A )  i^i  (  _I  |`  A ) )  =  (  _I  |`  A )
11 relfld 5198 . . . . . . 7  |-  ( Rel  (  _I  |`  A )  ->  U. U. (  _I  |`  A )  =  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) ) )
121, 11ax-mp 8 . . . . . 6  |-  U. U. (  _I  |`  A )  =  ( dom  (  _I  |`  A )  u. 
ran  (  _I  |`  A ) )
13 dmresi 5005 . . . . . . 7  |-  dom  (  _I  |`  A )  =  A
14 rnresi 5028 . . . . . . 7  |-  ran  (  _I  |`  A )  =  A
1513, 14uneq12i 3327 . . . . . 6  |-  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) )  =  ( A  u.  A
)
16 unidm 3318 . . . . . 6  |-  ( A  u.  A )  =  A
1712, 15, 163eqtrri 2308 . . . . 5  |-  A  = 
U. U. (  _I  |`  A )
1817reseq2i 4952 . . . 4  |-  (  _I  |`  A )  =  (  _I  |`  U. U. (  _I  |`  A ) )
199, 10, 183eqtri 2307 . . 3  |-  ( (  _I  |`  A )  i^i  `' (  _I  |`  A ) )  =  (  _I  |`  U. U. (  _I  |`  A ) )
201, 7, 193pm3.2i 1130 . 2  |-  ( Rel  (  _I  |`  A )  /\  ( (  _I  |`  A )  o.  (  _I  |`  A ) ) 
C_  (  _I  |`  A )  /\  ( (  _I  |`  A )  i^i  `' (  _I  |`  A ) )  =  (  _I  |`  U. U. (  _I  |`  A ) ) )
21 resiexg 4997 . . 3  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )
22 isps 14311 . . 3  |-  ( (  _I  |`  A )  e.  _V  ->  ( (  _I  |`  A )  e.  PosetRel  <->  ( Rel  (  _I  |`  A )  /\  ( (  _I  |`  A )  o.  (  _I  |`  A ) ) 
C_  (  _I  |`  A )  /\  ( (  _I  |`  A )  i^i  `' (  _I  |`  A ) )  =  (  _I  |`  U. U. (  _I  |`  A ) ) ) ) )
2321, 22syl 15 . 2  |-  ( A  e.  V  ->  (
(  _I  |`  A )  e.  PosetRel 
<->  ( Rel  (  _I  |`  A )  /\  (
(  _I  |`  A )  o.  (  _I  |`  A ) )  C_  (  _I  |`  A )  /\  (
(  _I  |`  A )  i^i  `' (  _I  |`  A ) )  =  (  _I  |`  U. U. (  _I  |`  A ) ) ) ) )
2420, 23mpbiri 224 1  |-  ( A  e.  V  ->  (  _I  |`  A )  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   U.cuni 3827    _I cid 4304   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694   -->wf 5251   -1-1-onto->wf1o 5254   PosetRelcps 14301
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-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-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-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-ps 14306
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