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Theorem imai 5210
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 5198 . 2  |-  (  _I  " A )  =  {
y  |  E. x
( x  e.  A  /\  <. x ,  y
>.  e.  _I  ) }
2 df-br 4205 . . . . . . . 8  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
3 vex 2951 . . . . . . . . 9  |-  y  e. 
_V
43ideq 5017 . . . . . . . 8  |-  ( x  _I  y  <->  x  =  y )
52, 4bitr3i 243 . . . . . . 7  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
65anbi2i 676 . . . . . 6  |-  ( ( x  e.  A  /\  <.
x ,  y >.  e.  _I  )  <->  ( x  e.  A  /\  x  =  y ) )
7 ancom 438 . . . . . 6  |-  ( ( x  e.  A  /\  x  =  y )  <->  ( x  =  y  /\  x  e.  A )
)
86, 7bitri 241 . . . . 5  |-  ( ( x  e.  A  /\  <.
x ,  y >.  e.  _I  )  <->  ( x  =  y  /\  x  e.  A ) )
98exbii 1592 . . . 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 2983 . . . 4  |-  ( E. x ( x  =  y  /\  x  e.  A )  <->  y  e.  A )
129, 11bitri 241 . . 3  |-  ( E. x ( x  e.  A  /\  <. x ,  y >.  e.  _I  ) 
<->  y  e.  A )
1312abbii 2547 . 2  |-  { y  |  E. x ( x  e.  A  /\  <.
x ,  y >.  e.  _I  ) }  =  { y  |  y  e.  A }
14 abid2 2552 . 2  |-  { y  |  y  e.  A }  =  A
151, 13, 143eqtri 2459 1  |-  (  _I  " A )  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   <.cop 3809   class class class wbr 4204    _I cid 4485   "cima 4873
This theorem is referenced by:  rnresi  5211  cnvresid  5515  ecidsn  6945  mbfid  19520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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