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Theorem imai 5199
Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
Assertion
Ref Expression
imai  |-  (  _I  " A )  =  A

Proof of Theorem imai
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfima3 5190 . 2  |-  (  _I  " A )  =  {
y  |  E. x
( x  e.  A  /\  <. x ,  y
>.  e.  _I  ) }
2 df-br 4424 . . . . . . . 8  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
3 vex 3083 . . . . . . . . 9  |-  y  e. 
_V
43ideq 5006 . . . . . . . 8  |-  ( x  _I  y  <->  x  =  y )
52, 4bitr3i 254 . . . . . . 7  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
65anbi2i 698 . . . . . 6  |-  ( ( x  e.  A  /\  <.
x ,  y >.  e.  _I  )  <->  ( x  e.  A  /\  x  =  y ) )
7 ancom 451 . . . . . 6  |-  ( ( x  e.  A  /\  x  =  y )  <->  ( x  =  y  /\  x  e.  A )
)
86, 7bitri 252 . . . . 5  |-  ( ( x  e.  A  /\  <.
x ,  y >.  e.  _I  )  <->  ( x  =  y  /\  x  e.  A ) )
98exbii 1712 . . . 4  |-  ( E. x ( x  e.  A  /\  <. x ,  y >.  e.  _I  ) 
<->  E. x ( x  =  y  /\  x  e.  A ) )
10 eleq1 2495 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
113, 10ceqsexv 3118 . . . 4  |-  ( E. x ( x  =  y  /\  x  e.  A )  <->  y  e.  A )
129, 11bitri 252 . . 3  |-  ( E. x ( x  e.  A  /\  <. x ,  y >.  e.  _I  ) 
<->  y  e.  A )
1312abbii 2551 . 2  |-  { y  |  E. x ( x  e.  A  /\  <.
x ,  y >.  e.  _I  ) }  =  { y  |  y  e.  A }
14 abid2 2558 . 2  |-  { y  |  y  e.  A }  =  A
151, 13, 143eqtri 2455 1  |-  (  _I  " A )  =  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407   <.cop 4004   class class class wbr 4423    _I cid 4763   "cima 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866
This theorem is referenced by:  rnresi  5200  cnvresid  5671  ecidsn  7423  mbfid  22590  frege131d  36326  frege110  36539  frege133  36562
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