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Theorem fnimage 25766
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 25765 . 2  |-  Fun Image R
2 vex 2951 . . . . . . . 8  |-  y  e. 
_V
3 vex 2951 . . . . . . . 8  |-  x  e. 
_V
42, 3brimage 25763 . . . . . . 7  |-  ( yImage
R x  <->  x  =  ( R " y ) )
5 eleq1 2495 . . . . . . . 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 1644 . . . . 5  |-  ( E. x  yImage R x  ->  ( R "
y )  e.  _V )
9 eqid 2435 . . . . . . 7  |-  ( R
" y )  =  ( R " y
)
10 brimageg 25764 . . . . . . . 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 4208 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  (
yImage R x  <->  yImage R
( R " y
) ) )
1413spcegv 3029 . . . . . 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 5059 . . . 4  |-  ( y  e.  dom Image R  <->  E. x  yImage R x )
18 imaeq2 5191 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2501 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
202, 19elab 3074 . . . 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 2432 . 2  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
23 df-fn 5449 . 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 1550    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   class class class wbr 4204   dom cdm 4870   "cima 4873   Fun wfun 5440    Fn wfn 5441  Imagecimage 25676
This theorem is referenced by:  imageval  25767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-eprel 4486  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-1st 6341  df-2nd 6342  df-symdif 25655  df-txp 25690  df-image 25700
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