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Theorem fnimage 24539
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 24538 . 2  |-  Fun Image R
2 vex 2804 . . . . . . . 8  |-  y  e. 
_V
3 vex 2804 . . . . . . . 8  |-  x  e. 
_V
42, 3brimage 24536 . . . . . . 7  |-  ( yImage
R x  <->  x  =  ( R " y ) )
5 eleq1 2356 . . . . . . . 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 1624 . . . . 5  |-  ( E. x  yImage R x  ->  ( R "
y )  e.  _V )
9 eqid 2296 . . . . . . 7  |-  ( R
" y )  =  ( R " y
)
10 brimageg 24537 . . . . . . . 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 4043 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  (
yImage R x  <->  yImage R
( R " y
) ) )
1413spcegv 2882 . . . . . 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 4892 . . . 4  |-  ( y  e.  dom Image R  <->  E. x  yImage R x )
18 imaeq2 5024 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2362 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
202, 19elab 2927 . . . 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 2293 . 2  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
23 df-fn 5274 . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   class class class wbr 4039   dom cdm 4705   "cima 4708   Fun wfun 5265    Fn wfn 5266  Imagecimage 24454
This theorem is referenced by:  imageval  24540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-image 24476
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