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Theorem iss 5152
Description: A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
iss  |-  ( A 
C_  _I  <->  A  =  (  _I  |`  dom  A ) )

Proof of Theorem iss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3348 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  _I  ) )
2 vex 2973 . . . . . . . . 9  |-  x  e. 
_V
3 vex 2973 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 5041 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
54a1i 11 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A ) )
61, 5jcad 533 . . . . . 6  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  _I  /\  x  e. 
dom  A ) ) )
7 df-br 4291 . . . . . . . . 9  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
83ideq 4990 . . . . . . . . 9  |-  ( x  _I  y  <->  x  =  y )
97, 8bitr3i 251 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
102eldm2 5036 . . . . . . . . . 10  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
11 opeq2 4058 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  <. x ,  x >.  =  <. x ,  y >. )
1211eleq1d 2507 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( <. x ,  x >.  e.  A  <->  <. x ,  y
>.  e.  A ) )
1312biimprcd 225 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  A  ->  ( x  =  y  ->  <. x ,  x >.  e.  A
) )
149, 13syl5bi 217 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  _I  ->  <. x ,  x >.  e.  A
) )
151, 14sylcom 29 . . . . . . . . . . 11  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  x >.  e.  A
) )
1615exlimdv 1690 . . . . . . . . . 10  |-  ( A 
C_  _I  ->  ( E. y <. x ,  y
>.  e.  A  ->  <. x ,  x >.  e.  A
) )
1710, 16syl5bi 217 . . . . . . . . 9  |-  ( A 
C_  _I  ->  ( x  e.  dom  A  ->  <. x ,  x >.  e.  A ) )
1812imbi2d 316 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  dom  A  ->  <. x ,  x >.  e.  A )  <->  ( x  e.  dom  A  ->  <. x ,  y >.  e.  A
) ) )
1917, 18syl5ibcom 220 . . . . . . . 8  |-  ( A 
C_  _I  ->  ( x  =  y  ->  (
x  e.  dom  A  -> 
<. x ,  y >.  e.  A ) ) )
209, 19syl5bi 217 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  _I  ->  ( x  e.  dom  A  ->  <. x ,  y >.  e.  A
) ) )
2120impd 431 . . . . . 6  |-  ( A 
C_  _I  ->  ( (
<. x ,  y >.  e.  _I  /\  x  e. 
dom  A )  ->  <. x ,  y >.  e.  A ) )
226, 21impbid 191 . . . . 5  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) ) )
233opelres 5114 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  dom  A
)  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) )
2422, 23syl6bbr 263 . . . 4  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  (  _I  |`  dom  A
) ) )
2524alrimivv 1686 . . 3  |-  ( A 
C_  _I  ->  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  (  _I  |`  dom  A
) ) )
26 reli 4965 . . . . 5  |-  Rel  _I
27 relss 4925 . . . . 5  |-  ( A 
C_  _I  ->  ( Rel 
_I  ->  Rel  A )
)
2826, 27mpi 17 . . . 4  |-  ( A 
C_  _I  ->  Rel  A
)
29 relres 5136 . . . 4  |-  Rel  (  _I  |`  dom  A )
30 eqrel 4927 . . . 4  |-  ( ( Rel  A  /\  Rel  (  _I  |`  dom  A
) )  ->  ( A  =  (  _I  |` 
dom  A )  <->  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  (  _I  |`  dom  A
) ) ) )
3128, 29, 30sylancl 662 . . 3  |-  ( A 
C_  _I  ->  ( A  =  (  _I  |`  dom  A
)  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  (  _I  |`  dom  A ) ) ) )
3225, 31mpbird 232 . 2  |-  ( A 
C_  _I  ->  A  =  (  _I  |`  dom  A
) )
33 resss 5132 . . 3  |-  (  _I  |`  dom  A )  C_  _I
34 sseq1 3375 . . 3  |-  ( A  =  (  _I  |`  dom  A
)  ->  ( A  C_  _I  <->  (  _I  |`  dom  A
)  C_  _I  )
)
3533, 34mpbiri 233 . 2  |-  ( A  =  (  _I  |`  dom  A
)  ->  A  C_  _I  )
3632, 35impbii 188 1  |-  ( A 
C_  _I  <->  A  =  (  _I  |`  dom  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756    C_ wss 3326   <.cop 3881   class class class wbr 4290    _I cid 4629   dom cdm 4838    |` cres 4840   Rel wrel 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-dm 4848  df-res 4850
This theorem is referenced by:  funcocnv2  5663  trust  19802
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