MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iss Structured version   Visualization version   Unicode version

Theorem iss 5158
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 3412 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  _I  ) )
2 vex 3034 . . . . . . . . 9  |-  x  e. 
_V
3 vex 3034 . . . . . . . . 9  |-  y  e. 
_V
42, 3opeldm 5044 . . . . . . . 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 542 . . . . . 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 4992 . . . . . . . . 9  |-  ( x  _I  y  <->  x  =  y )
97, 8bitr3i 259 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
102eldm2 5038 . . . . . . . . . 10  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
11 opeq2 4159 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  <. x ,  x >.  =  <. x ,  y >. )
1211eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  ( <. x ,  x >.  e.  A  <->  <. x ,  y
>.  e.  A ) )
1312biimprcd 233 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  A  ->  ( x  =  y  ->  <. x ,  x >.  e.  A
) )
149, 13syl5bi 225 . . . . . . . . . . . 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 1787 . . . . . . . . . 10  |-  ( A 
C_  _I  ->  ( E. y <. x ,  y
>.  e.  A  ->  <. x ,  x >.  e.  A
) )
1710, 16syl5bi 225 . . . . . . . . 9  |-  ( A 
C_  _I  ->  ( x  e.  dom  A  ->  <. x ,  x >.  e.  A ) )
1812imbi2d 323 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  dom  A  ->  <. x ,  x >.  e.  A )  <->  ( x  e.  dom  A  ->  <. x ,  y >.  e.  A
) ) )
1917, 18syl5ibcom 228 . . . . . . . 8  |-  ( A 
C_  _I  ->  ( x  =  y  ->  (
x  e.  dom  A  -> 
<. x ,  y >.  e.  A ) ) )
209, 19syl5bi 225 . . . . . . 7  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  _I  ->  ( x  e.  dom  A  ->  <. x ,  y >.  e.  A
) ) )
2120impd 438 . . . . . 6  |-  ( A 
C_  _I  ->  ( (
<. x ,  y >.  e.  _I  /\  x  e. 
dom  A )  ->  <. x ,  y >.  e.  A ) )
226, 21impbid 195 . . . . 5  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) ) )
233opelres 5116 . . . . 5  |-  ( <.
x ,  y >.  e.  (  _I  |`  dom  A
)  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  dom  A ) )
2422, 23syl6bbr 271 . . . 4  |-  ( A 
C_  _I  ->  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  (  _I  |`  dom  A
) ) )
2524alrimivv 1782 . . 3  |-  ( A 
C_  _I  ->  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  (  _I  |`  dom  A
) ) )
26 reli 4967 . . . . 5  |-  Rel  _I
27 relss 4927 . . . . 5  |-  ( A 
C_  _I  ->  ( Rel 
_I  ->  Rel  A )
)
2826, 27mpi 20 . . . 4  |-  ( A 
C_  _I  ->  Rel  A
)
29 relres 5138 . . . 4  |-  Rel  (  _I  |`  dom  A )
30 eqrel 4929 . . . 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 675 . . 3  |-  ( A 
C_  _I  ->  ( A  =  (  _I  |`  dom  A
)  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  (  _I  |`  dom  A ) ) ) )
3225, 31mpbird 240 . 2  |-  ( A 
C_  _I  ->  A  =  (  _I  |`  dom  A
) )
33 resss 5134 . . 3  |-  (  _I  |`  dom  A )  C_  _I
34 sseq1 3439 . . 3  |-  ( A  =  (  _I  |`  dom  A
)  ->  ( A  C_  _I  <->  (  _I  |`  dom  A
)  C_  _I  )
)
3533, 34mpbiri 241 . 2  |-  ( A  =  (  _I  |`  dom  A
)  ->  A  C_  _I  )
3632, 35impbii 192 1  |-  ( A 
C_  _I  <->  A  =  (  _I  |`  dom  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904    C_ wss 3390   <.cop 3965   class class class wbr 4395    _I cid 4749   dom cdm 4839    |` cres 4841   Rel wrel 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-dm 4849  df-res 4851
This theorem is referenced by:  funcocnv2  5852  trust  21322
  Copyright terms: Public domain W3C validator