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Theorem imai 5026
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 5014 . 2  |-  (  _I  " A )  =  {
y  |  E. x
( x  e.  A  /\  <. x ,  y
>.  e.  _I  ) }
2 df-br 4025 . . . . . . . 8  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
3 vex 2792 . . . . . . . . 9  |-  y  e. 
_V
43ideq 4835 . . . . . . . 8  |-  ( x  _I  y  <->  x  =  y )
52, 4bitr3i 242 . . . . . . 7  |-  ( <.
x ,  y >.  e.  _I  <->  x  =  y
)
65anbi2i 675 . . . . . 6  |-  ( ( x  e.  A  /\  <.
x ,  y >.  e.  _I  )  <->  ( x  e.  A  /\  x  =  y ) )
7 ancom 437 . . . . . 6  |-  ( ( x  e.  A  /\  x  =  y )  <->  ( x  =  y  /\  x  e.  A )
)
86, 7bitri 240 . . . . 5  |-  ( ( x  e.  A  /\  <.
x ,  y >.  e.  _I  )  <->  ( x  =  y  /\  x  e.  A ) )
98exbii 1569 . . . 4  |-  ( E. x ( x  e.  A  /\  <. x ,  y >.  e.  _I  ) 
<->  E. x ( x  =  y  /\  x  e.  A ) )
10 eleq1 2344 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
113, 10ceqsexv 2824 . . . 4  |-  ( E. x ( x  =  y  /\  x  e.  A )  <->  y  e.  A )
129, 11bitri 240 . . 3  |-  ( E. x ( x  e.  A  /\  <. x ,  y >.  e.  _I  ) 
<->  y  e.  A )
1312abbii 2396 . 2  |-  { y  |  E. x ( x  e.  A  /\  <.
x ,  y >.  e.  _I  ) }  =  { y  |  y  e.  A }
14 abid2 2401 . 2  |-  { y  |  y  e.  A }  =  A
151, 13, 143eqtri 2308 1  |-  (  _I  " A )  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   {cab 2270   <.cop 3644   class class class wbr 4024    _I cid 4303   "cima 4691
This theorem is referenced by:  rnresi  5027  cnvresid  5288  ecidsn  6704  mbfid  18987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701
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