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Theorem dispos 25390
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 4999 . . 3  |-  Rel  (  _I  |`  A )
2 f1oi 5527 . . . . . 6  |-  (  _I  |`  A ) : A -1-1-onto-> A
3 f1of 5488 . . . . . 6  |-  ( (  _I  |`  A ) : A -1-1-onto-> A  ->  (  _I  |`  A ) : A --> A )
42, 3ax-mp 8 . . . . 5  |-  (  _I  |`  A ) : A --> A
5 fcoi1 5431 . . . . 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 3245 . . 3  |-  ( (  _I  |`  A )  o.  (  _I  |`  A ) )  C_  (  _I  |`  A )
8 cnvresid 5338 . . . . 5  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
98ineq2i 3380 . . . 4  |-  ( (  _I  |`  A )  i^i  `' (  _I  |`  A ) )  =  ( (  _I  |`  A )  i^i  (  _I  |`  A ) )
10 inidm 3391 . . . 4  |-  ( (  _I  |`  A )  i^i  (  _I  |`  A ) )  =  (  _I  |`  A )
11 relfld 5214 . . . . . . 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 5021 . . . . . . 7  |-  dom  (  _I  |`  A )  =  A
14 rnresi 5044 . . . . . . 7  |-  ran  (  _I  |`  A )  =  A
1513, 14uneq12i 3340 . . . . . 6  |-  ( dom  (  _I  |`  A )  u.  ran  (  _I  |`  A ) )  =  ( A  u.  A
)
16 unidm 3331 . . . . . 6  |-  ( A  u.  A )  =  A
1712, 15, 163eqtrri 2321 . . . . 5  |-  A  = 
U. U. (  _I  |`  A )
1817reseq2i 4968 . . . 4  |-  (  _I  |`  A )  =  (  _I  |`  U. U. (  _I  |`  A ) )
199, 10, 183eqtri 2320 . . 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 5013 . . 3  |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
_V )
22 isps 14327 . . 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 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   U.cuni 3843    _I cid 4320   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707    o. ccom 4709   Rel wrel 4710   -->wf 5267   -1-1-onto->wf1o 5270   PosetRelcps 14317
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-pw 3640  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  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-ps 14322
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