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Theorem restidsing 25179
Description: Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.)
Assertion
Ref Expression
restidsing  |-  (  _I  |`  { A } )  =  ( { A }  X.  { A }
)

Proof of Theorem restidsing
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4999 . 2  |-  Rel  (  _I  |`  { A }
)
2 relxp 4810 . 2  |-  Rel  ( { A }  X.  { A } )
3 df-br 4040 . . . . . 6  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
43bicomi 193 . . . . 5  |-  ( <.
x ,  y >.  e.  _I  <->  x  _I  y
)
54anbi1i 676 . . . 4  |-  ( (
<. x ,  y >.  e.  _I  /\  x  e. 
{ A } )  <-> 
( x  _I  y  /\  x  e.  { A } ) )
6 simpr 447 . . . . . 6  |-  ( ( x  _I  y  /\  x  e.  { A } )  ->  x  e.  { A } )
7 elsn 3668 . . . . . . . 8  |-  ( x  e.  { A }  <->  x  =  A )
8 vex 2804 . . . . . . . . . 10  |-  y  e. 
_V
9 ideqg 4851 . . . . . . . . . . 11  |-  ( y  e.  _V  ->  (
x  _I  y  <->  x  =  y ) )
109biimpd 198 . . . . . . . . . 10  |-  ( y  e.  _V  ->  (
x  _I  y  ->  x  =  y )
)
118, 10ax-mp 8 . . . . . . . . 9  |-  ( x  _I  y  ->  x  =  y )
12 eqtr2 2314 . . . . . . . . . . 11  |-  ( ( x  =  y  /\  x  =  A )  ->  y  =  A )
1312ex 423 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  =  A  -> 
y  =  A ) )
14 elsn 3668 . . . . . . . . . 10  |-  ( y  e.  { A }  <->  y  =  A )
1513, 14syl6ibr 218 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  =  A  -> 
y  e.  { A } ) )
1611, 15syl 15 . . . . . . . 8  |-  ( x  _I  y  ->  (
x  =  A  -> 
y  e.  { A } ) )
177, 16syl5bi 208 . . . . . . 7  |-  ( x  _I  y  ->  (
x  e.  { A }  ->  y  e.  { A } ) )
1817imp 418 . . . . . 6  |-  ( ( x  _I  y  /\  x  e.  { A } )  ->  y  e.  { A } )
196, 18jca 518 . . . . 5  |-  ( ( x  _I  y  /\  x  e.  { A } )  ->  (
x  e.  { A }  /\  y  e.  { A } ) )
20 eqtr3 2315 . . . . . . . . . . . 12  |-  ( ( y  =  A  /\  x  =  A )  ->  y  =  x )
218ideq 4852 . . . . . . . . . . . . 13  |-  ( x  _I  y  <->  x  =  y )
22 equcom 1665 . . . . . . . . . . . . 13  |-  ( x  =  y  <->  y  =  x )
2321, 22bitri 240 . . . . . . . . . . . 12  |-  ( x  _I  y  <->  y  =  x )
2420, 23sylibr 203 . . . . . . . . . . 11  |-  ( ( y  =  A  /\  x  =  A )  ->  x  _I  y )
2524ex 423 . . . . . . . . . 10  |-  ( y  =  A  ->  (
x  =  A  ->  x  _I  y )
)
2614, 25sylbi 187 . . . . . . . . 9  |-  ( y  e.  { A }  ->  ( x  =  A  ->  x  _I  y
) )
2726com12 27 . . . . . . . 8  |-  ( x  =  A  ->  (
y  e.  { A }  ->  x  _I  y
) )
287, 27sylbi 187 . . . . . . 7  |-  ( x  e.  { A }  ->  ( y  e.  { A }  ->  x  _I  y ) )
2928imp 418 . . . . . 6  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  x  _I  y )
30 simpl 443 . . . . . 6  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  x  e.  { A } )
3129, 30jca 518 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x  _I  y  /\  x  e.  { A } ) )
3219, 31impbii 180 . . . 4  |-  ( ( x  _I  y  /\  x  e.  { A } )  <->  ( x  e.  { A }  /\  y  e.  { A } ) )
335, 32bitri 240 . . 3  |-  ( (
<. x ,  y >.  e.  _I  /\  x  e. 
{ A } )  <-> 
( x  e.  { A }  /\  y  e.  { A } ) )
348opelres 4976 . . 3  |-  ( <.
x ,  y >.  e.  (  _I  |`  { A } )  <->  ( <. x ,  y >.  e.  _I  /\  x  e.  { A } ) )
35 opelxp 4735 . . 3  |-  ( <.
x ,  y >.  e.  ( { A }  X.  { A } )  <-> 
( x  e.  { A }  /\  y  e.  { A } ) )
3633, 34, 353bitr4i 268 . 2  |-  ( <.
x ,  y >.  e.  (  _I  |`  { A } )  <->  <. x ,  y >.  e.  ( { A }  X.  { A } ) )
371, 2, 36eqrelriiv 4797 1  |-  (  _I  |`  { A } )  =  ( { A }  X.  { A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   class class class wbr 4039    _I cid 4320    X. cxp 4703    |` cres 4707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-res 4717
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