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Theorem imai 5181
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 5169 . 2  |-  (  _I  " A )  =  {
y  |  E. x
( x  e.  A  /\  <. x ,  y
>.  e.  _I  ) }
2 df-br 4177 . . . . . . . 8  |-  ( x  _I  y  <->  <. x ,  y >.  e.  _I  )
3 vex 2923 . . . . . . . . 9  |-  y  e. 
_V
43ideq 4988 . . . . . . . 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 1589 . . . 4  |-  ( E. x ( x  e.  A  /\  <. x ,  y >.  e.  _I  ) 
<->  E. x ( x  =  y  /\  x  e.  A ) )
10 eleq1 2468 . . . . 5  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
113, 10ceqsexv 2955 . . . 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 2520 . 2  |-  { y  |  E. x ( x  e.  A  /\  <.
x ,  y >.  e.  _I  ) }  =  { y  |  y  e.  A }
14 abid2 2525 . 2  |-  { y  |  y  e.  A }  =  A
151, 13, 143eqtri 2432 1  |-  (  _I  " A )  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2394   <.cop 3781   class class class wbr 4176    _I cid 4457   "cima 4844
This theorem is referenced by:  rnresi  5182  cnvresid  5486  ecidsn  6916  mbfid  19485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854
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