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Theorem fnimage 25492
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 25491 . 2  |-  Fun Image R
2 vex 2902 . . . . . . . 8  |-  y  e. 
_V
3 vex 2902 . . . . . . . 8  |-  x  e. 
_V
42, 3brimage 25489 . . . . . . 7  |-  ( yImage
R x  <->  x  =  ( R " y ) )
5 eleq1 2447 . . . . . . . 8  |-  ( x  =  ( R "
y )  ->  (
x  e.  _V  <->  ( R " y )  e.  _V ) )
63, 5mpbii 203 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  ( R " y )  e. 
_V )
74, 6sylbi 188 . . . . . 6  |-  ( yImage
R x  ->  ( R " y )  e. 
_V )
87exlimiv 1641 . . . . 5  |-  ( E. x  yImage R x  ->  ( R "
y )  e.  _V )
9 eqid 2387 . . . . . . 7  |-  ( R
" y )  =  ( R " y
)
10 brimageg 25490 . . . . . . . 8  |-  ( ( y  e.  _V  /\  ( R " y )  e.  _V )  -> 
( yImage R ( R " y )  <-> 
( R " y
)  =  ( R
" y ) ) )
112, 10mpan 652 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
yImage R ( R
" y )  <->  ( R " y )  =  ( R " y ) ) )
129, 11mpbiri 225 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  yImage R ( R "
y ) )
13 breq2 4157 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  (
yImage R x  <->  yImage R
( R " y
) ) )
1413spcegv 2980 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  (
yImage R ( R
" y )  ->  E. x  yImage R x ) )
1512, 14mpd 15 . . . . 5  |-  ( ( R " y )  e.  _V  ->  E. x  yImage R x )
168, 15impbii 181 . . . 4  |-  ( E. x  yImage R x  <-> 
( R " y
)  e.  _V )
172eldm 5007 . . . 4  |-  ( y  e.  dom Image R  <->  E. x  yImage R x )
18 imaeq2 5139 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2453 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
202, 19elab 3025 . . . 4  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  <->  ( R " y )  e.  _V )
2116, 17, 203bitr4i 269 . . 3  |-  ( y  e.  dom Image R  <->  y  e.  { x  |  ( R
" x )  e. 
_V } )
2221eqriv 2384 . 2  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
23 df-fn 5397 . 2  |-  (Image R  Fn  { x  |  ( R " x )  e.  _V }  <->  ( Fun Image R  /\  dom Image R  =  {
x  |  ( R
" x )  e. 
_V } ) )
241, 22, 23mpbir2an 887 1  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373   _Vcvv 2899   class class class wbr 4153   dom cdm 4818   "cima 4821   Fun wfun 5388    Fn wfn 5389  Imagecimage 25407
This theorem is referenced by:  imageval  25493
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-eprel 4435  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-1st 6288  df-2nd 6289  df-symdif 25386  df-txp 25419  df-image 25429
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