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Theorem iss 5141
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 3436 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  _I  ) )
2 vex 3062 . . . . . . . . 9  |-  x  e. 
_V
3 vex 3062 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 5027 . . . . . . . 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 531 . . . . . 6  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  _I  /\  x  e. 
dom  A ) ) )
7 df-br 4396 . . . . . . . . 9  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
83ideq 4976 . . . . . . . . 9  |-  ( x  _I  y  <->  x  =  y )
97, 8bitr3i 251 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
102eldm2 5022 . . . . . . . . . 10  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
11 opeq2 4160 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  <. x ,  x >.  =  <. x ,  y >. )
1211eleq1d 2471 . . . . . . . . . . . . . 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 27 . . . . . . . . . . 11  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  x >.  e.  A
) )
1615exlimdv 1745 . . . . . . . . . 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 314 . . . . . . . . 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 429 . . . . . 6  |-  ( A 
C_  _I  ->  ( (
<. x ,  y >.  e.  _I  /\  x  e. 
dom  A )  ->  <. x ,  y >.  e.  A ) )
226, 21impbid 190 . . . . 5  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) ) )
233opelres 5099 . . . . 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 1741 . . 3  |-  ( A 
C_  _I  ->  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  (  _I  |`  dom  A
) ) )
26 reli 4951 . . . . 5  |-  Rel  _I
27 relss 4911 . . . . 5  |-  ( A 
C_  _I  ->  ( Rel 
_I  ->  Rel  A )
)
2826, 27mpi 20 . . . 4  |-  ( A 
C_  _I  ->  Rel  A
)
29 relres 5121 . . . 4  |-  Rel  (  _I  |`  dom  A )
30 eqrel 4913 . . . 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 660 . . 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 5117 . . 3  |-  (  _I  |`  dom  A )  C_  _I
34 sseq1 3463 . . 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 187 1  |-  ( A 
C_  _I  <->  A  =  (  _I  |`  dom  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405   E.wex 1633    e. wcel 1842    C_ wss 3414   <.cop 3978   class class class wbr 4395    _I cid 4733   dom cdm 4823    |` cres 4825   Rel wrel 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-dm 4833  df-res 4835
This theorem is referenced by:  funcocnv2  5823  trust  21024
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