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Theorem iss 5080
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 3250 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  _I  ) )
2 vex 2867 . . . . . . . . 9  |-  x  e. 
_V
3 vex 2867 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 4964 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
54a1i 10 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A ) )
61, 5jcad 519 . . . . . 6  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  _I  /\  x  e. 
dom  A ) ) )
7 df-br 4105 . . . . . . . . 9  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
83ideq 4918 . . . . . . . . 9  |-  ( x  _I  y  <->  x  =  y )
97, 8bitr3i 242 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
102eldm2 4959 . . . . . . . . . 10  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
11 opeq2 3878 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  <. x ,  x >.  =  <. x ,  y >. )
1211eleq1d 2424 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( <. x ,  x >.  e.  A  <->  <. x ,  y
>.  e.  A ) )
1312biimprcd 216 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  A  ->  ( x  =  y  ->  <. x ,  x >.  e.  A
) )
149, 13syl5bi 208 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  A  ->  ( <.
x ,  y >.  e.  _I  ->  <. x ,  x >.  e.  A
) )
151, 14sylcom 25 . . . . . . . . . . 11  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  x >.  e.  A
) )
1615exlimdv 1636 . . . . . . . . . 10  |-  ( A 
C_  _I  ->  ( E. y <. x ,  y
>.  e.  A  ->  <. x ,  x >.  e.  A
) )
1710, 16syl5bi 208 . . . . . . . . 9  |-  ( A 
C_  _I  ->  ( x  e.  dom  A  ->  <. x ,  x >.  e.  A ) )
1812imbi2d 307 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  dom  A  ->  <. x ,  x >.  e.  A )  <->  ( x  e.  dom  A  ->  <. x ,  y >.  e.  A
) ) )
1917, 18syl5ibcom 211 . . . . . . . 8  |-  ( A 
C_  _I  ->  ( x  =  y  ->  (
x  e.  dom  A  -> 
<. x ,  y >.  e.  A ) ) )
209, 19syl5bi 208 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  _I  ->  ( x  e.  dom  A  ->  <. x ,  y >.  e.  A
) ) )
2120imp3a 420 . . . . . 6  |-  ( A 
C_  _I  ->  ( (
<. x ,  y >.  e.  _I  /\  x  e. 
dom  A )  ->  <. x ,  y >.  e.  A ) )
226, 21impbid 183 . . . . 5  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) ) )
233opelres 5042 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  dom  A
)  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) )
2422, 23syl6bbr 254 . . . 4  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  (  _I  |`  dom  A
) ) )
2524alrimivv 1632 . . 3  |-  ( A 
C_  _I  ->  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  (  _I  |`  dom  A
) ) )
26 reli 4895 . . . . 5  |-  Rel  _I
27 relss 4857 . . . . 5  |-  ( A 
C_  _I  ->  ( Rel 
_I  ->  Rel  A )
)
2826, 27mpi 16 . . . 4  |-  ( A 
C_  _I  ->  Rel  A
)
29 relres 5065 . . . 4  |-  Rel  (  _I  |`  dom  A )
30 eqrel 4859 . . . 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 643 . . 3  |-  ( A 
C_  _I  ->  ( A  =  (  _I  |`  dom  A
)  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  (  _I  |`  dom  A ) ) ) )
3225, 31mpbird 223 . 2  |-  ( A 
C_  _I  ->  A  =  (  _I  |`  dom  A
) )
33 resss 5061 . . 3  |-  (  _I  |`  dom  A )  C_  _I
34 sseq1 3275 . . 3  |-  ( A  =  (  _I  |`  dom  A
)  ->  ( A  C_  _I  <->  (  _I  |`  dom  A
)  C_  _I  )
)
3533, 34mpbiri 224 . 2  |-  ( A  =  (  _I  |`  dom  A
)  ->  A  C_  _I  )
3632, 35impbii 180 1  |-  ( A 
C_  _I  <->  A  =  (  _I  |`  dom  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1540   E.wex 1541    = wceq 1642    e. wcel 1710    C_ wss 3228   <.cop 3719   class class class wbr 4104    _I cid 4386   dom cdm 4771    |` cres 4773   Rel wrel 4776
This theorem is referenced by:  funcocnv2  5581  trust  23533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-dm 4781  df-res 4783
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