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Theorem fnimage 24468
Description: Image R is a function over the set-like portion of  R. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fnimage  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
Distinct variable group:    x, R

Proof of Theorem fnimage
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 funimage 24467 . 2  |-  Fun Image R
2 vex 2791 . . . . . . . 8  |-  y  e. 
_V
3 vex 2791 . . . . . . . 8  |-  x  e. 
_V
42, 3brimage 24465 . . . . . . 7  |-  ( yImage
R x  <->  x  =  ( R " y ) )
5 eleq1 2343 . . . . . . . 8  |-  ( x  =  ( R "
y )  ->  (
x  e.  _V  <->  ( R " y )  e.  _V ) )
63, 5mpbii 202 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  ( R " y )  e. 
_V )
74, 6sylbi 187 . . . . . 6  |-  ( yImage
R x  ->  ( R " y )  e. 
_V )
87exlimiv 1666 . . . . 5  |-  ( E. x  yImage R x  ->  ( R "
y )  e.  _V )
9 eqid 2283 . . . . . . 7  |-  ( R
" y )  =  ( R " y
)
10 brimageg 24466 . . . . . . . 8  |-  ( ( y  e.  _V  /\  ( R " y )  e.  _V )  -> 
( yImage R ( R " y )  <-> 
( R " y
)  =  ( R
" y ) ) )
112, 10mpan 651 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
yImage R ( R
" y )  <->  ( R " y )  =  ( R " y ) ) )
129, 11mpbiri 224 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  yImage R ( R "
y ) )
13 breq2 4027 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  (
yImage R x  <->  yImage R
( R " y
) ) )
1413spcegv 2869 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  (
yImage R ( R
" y )  ->  E. x  yImage R x ) )
1512, 14mpd 14 . . . . 5  |-  ( ( R " y )  e.  _V  ->  E. x  yImage R x )
168, 15impbii 180 . . . 4  |-  ( E. x  yImage R x  <-> 
( R " y
)  e.  _V )
172eldm 4876 . . . 4  |-  ( y  e.  dom Image R  <->  E. x  yImage R x )
18 imaeq2 5008 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2349 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
202, 19elab 2914 . . . 4  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  <->  ( R " y )  e.  _V )
2116, 17, 203bitr4i 268 . . 3  |-  ( y  e.  dom Image R  <->  y  e.  { x  |  ( R
" x )  e. 
_V } )
2221eqriv 2280 . 2  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
23 df-fn 5258 . 2  |-  (Image R  Fn  { x  |  ( R " x )  e.  _V }  <->  ( Fun Image R  /\  dom Image R  =  {
x  |  ( R
" x )  e. 
_V } ) )
241, 22, 23mpbir2an 886 1  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   class class class wbr 4023   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250  Imagecimage 24383
This theorem is referenced by:  imageval  24469
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-image 24405
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