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Theorem iss 4684
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 2994 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  _I  ) )
2 vex 2605 . . . . . . . . 9  |-  x  e. 
_V
3 vex 2605 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 4566 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
54a1i 9 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A ) )
61, 5jcad 301 . . . . . 6  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  _I  /\  x  e. 
dom  A ) ) )
7 df-br 3794 . . . . . . . . 9  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
83ideq 4516 . . . . . . . . 9  |-  ( x  _I  y  <->  x  =  y )
97, 8bitr3i 184 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
102eldm2 4561 . . . . . . . . . 10  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
11 opeq2 3579 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  <. x ,  x >.  =  <. x ,  y >. )
1211eleq1d 2148 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( <. x ,  x >.  e.  A  <->  <. x ,  y
>.  e.  A ) )
1312biimprcd 158 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  A  ->  ( x  =  y  ->  <. x ,  x >.  e.  A
) )
149, 13syl5bi 150 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  _I  ->  <. x ,  x >.  e.  A
) )
151, 14sylcom 28 . . . . . . . . . . 11  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  x >.  e.  A
) )
1615exlimdv 1741 . . . . . . . . . 10  |-  ( A 
C_  _I  ->  ( E. y <. x ,  y
>.  e.  A  ->  <. x ,  x >.  e.  A
) )
1710, 16syl5bi 150 . . . . . . . . 9  |-  ( A 
C_  _I  ->  ( x  e.  dom  A  ->  <. x ,  x >.  e.  A ) )
1812imbi2d 228 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  dom  A  ->  <. x ,  x >.  e.  A )  <->  ( x  e.  dom  A  ->  <. x ,  y >.  e.  A
) ) )
1917, 18syl5ibcom 153 . . . . . . . 8  |-  ( A 
C_  _I  ->  ( x  =  y  ->  (
x  e.  dom  A  -> 
<. x ,  y >.  e.  A ) ) )
209, 19syl5bi 150 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  _I  ->  ( x  e.  dom  A  ->  <. x ,  y >.  e.  A
) ) )
2120impd 251 . . . . . 6  |-  ( A 
C_  _I  ->  ( (
<. x ,  y >.  e.  _I  /\  x  e. 
dom  A )  ->  <. x ,  y >.  e.  A ) )
226, 21impbid 127 . . . . 5  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) ) )
233opelres 4645 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  dom  A
)  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) )
2422, 23syl6bbr 196 . . . 4  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  (  _I  |`  dom  A
) ) )
2524alrimivv 1797 . . 3  |-  ( A 
C_  _I  ->  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  (  _I  |`  dom  A
) ) )
26 reli 4493 . . . . 5  |-  Rel  _I
27 relss 4453 . . . . 5  |-  ( A 
C_  _I  ->  ( Rel 
_I  ->  Rel  A )
)
2826, 27mpi 15 . . . 4  |-  ( A 
C_  _I  ->  Rel  A
)
29 relres 4667 . . . 4  |-  Rel  (  _I  |`  dom  A )
30 eqrel 4455 . . . 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 404 . . 3  |-  ( A 
C_  _I  ->  ( A  =  (  _I  |`  dom  A
)  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  (  _I  |`  dom  A ) ) ) )
3225, 31mpbird 165 . 2  |-  ( A 
C_  _I  ->  A  =  (  _I  |`  dom  A
) )
33 resss 4663 . . 3  |-  (  _I  |`  dom  A )  C_  _I
34 sseq1 3021 . . 3  |-  ( A  =  (  _I  |`  dom  A
)  ->  ( A  C_  _I  <->  (  _I  |`  dom  A
)  C_  _I  )
)
3533, 34mpbiri 166 . 2  |-  ( A  =  (  _I  |`  dom  A
)  ->  A  C_  _I  )
3632, 35impbii 124 1  |-  ( A 
C_  _I  <->  A  =  (  _I  |`  dom  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434    C_ wss 2974   <.cop 3409   class class class wbr 3793    _I cid 4051   dom cdm 4371    |` cres 4373   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-dm 4381  df-res 4383
This theorem is referenced by:  funcocnv2  5182
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