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Theorem fnimage 29807
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 29806 . 2  |-  Fun Image R
2 vex 3109 . . . . . . . 8  |-  y  e. 
_V
3 vex 3109 . . . . . . . 8  |-  x  e. 
_V
42, 3brimage 29804 . . . . . . 7  |-  ( yImage
R x  <->  x  =  ( R " y ) )
5 eqvisset 3114 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  ( R " y )  e. 
_V )
64, 5sylbi 195 . . . . . 6  |-  ( yImage
R x  ->  ( R " y )  e. 
_V )
76exlimiv 1727 . . . . 5  |-  ( E. x  yImage R x  ->  ( R "
y )  e.  _V )
8 eqid 2454 . . . . . . 7  |-  ( R
" y )  =  ( R " y
)
9 brimageg 29805 . . . . . . . 8  |-  ( ( y  e.  _V  /\  ( R " y )  e.  _V )  -> 
( yImage R ( R " y )  <-> 
( R " y
)  =  ( R
" y ) ) )
102, 9mpan 668 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
yImage R ( R
" y )  <->  ( R " y )  =  ( R " y ) ) )
118, 10mpbiri 233 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  yImage R ( R "
y ) )
12 breq2 4443 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  (
yImage R x  <->  yImage R
( R " y
) ) )
1312spcegv 3192 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  (
yImage R ( R
" y )  ->  E. x  yImage R x ) )
1411, 13mpd 15 . . . . 5  |-  ( ( R " y )  e.  _V  ->  E. x  yImage R x )
157, 14impbii 188 . . . 4  |-  ( E. x  yImage R x  <-> 
( R " y
)  e.  _V )
162eldm 5189 . . . 4  |-  ( y  e.  dom Image R  <->  E. x  yImage R x )
17 imaeq2 5321 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1817eleq1d 2523 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
192, 18elab 3243 . . . 4  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  <->  ( R " y )  e.  _V )
2015, 16, 193bitr4i 277 . . 3  |-  ( y  e.  dom Image R  <->  y  e.  { x  |  ( R
" x )  e. 
_V } )
2120eqriv 2450 . 2  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
22 df-fn 5573 . 2  |-  (Image R  Fn  { x  |  ( R " x )  e.  _V }  <->  ( Fun Image R  /\  dom Image R  =  {
x  |  ( R
" x )  e. 
_V } ) )
231, 21, 22mpbir2an 918 1  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   _Vcvv 3106   class class class wbr 4439   dom cdm 4988   "cima 4991   Fun wfun 5564    Fn wfn 5565  Imagecimage 29717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-symdif 3715  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-eprel 4780  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773  df-2nd 6774  df-txp 29731  df-image 29741
This theorem is referenced by:  imageval  29808
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