Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnimage Structured version   Unicode version

Theorem fnimage 28099
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 28098 . 2  |-  Fun Image R
2 vex 3075 . . . . . . . 8  |-  y  e. 
_V
3 vex 3075 . . . . . . . 8  |-  x  e. 
_V
42, 3brimage 28096 . . . . . . 7  |-  ( yImage
R x  <->  x  =  ( R " y ) )
5 eqvisset 3080 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  ( R " y )  e. 
_V )
64, 5sylbi 195 . . . . . 6  |-  ( yImage
R x  ->  ( R " y )  e. 
_V )
76exlimiv 1689 . . . . 5  |-  ( E. x  yImage R x  ->  ( R "
y )  e.  _V )
8 eqid 2452 . . . . . . 7  |-  ( R
" y )  =  ( R " y
)
9 brimageg 28097 . . . . . . . 8  |-  ( ( y  e.  _V  /\  ( R " y )  e.  _V )  -> 
( yImage R ( R " y )  <-> 
( R " y
)  =  ( R
" y ) ) )
102, 9mpan 670 . . . . . . 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 4399 . . . . . . 7  |-  ( x  =  ( R "
y )  ->  (
yImage R x  <->  yImage R
( R " y
) ) )
1312spcegv 3158 . . . . . 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 5140 . . . 4  |-  ( y  e.  dom Image R  <->  E. x  yImage R x )
17 imaeq2 5268 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1817eleq1d 2521 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
192, 18elab 3207 . . . 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 2448 . 2  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
22 df-fn 5524 . 2  |-  (Image R  Fn  { x  |  ( R " x )  e.  _V }  <->  ( Fun Image R  /\  dom Image R  =  {
x  |  ( R
" x )  e. 
_V } ) )
231, 21, 22mpbir2an 911 1  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2437   _Vcvv 3072   class class class wbr 4395   dom cdm 4943   "cima 4946   Fun wfun 5515    Fn wfn 5516  Imagecimage 28009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-eprel 4735  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529  df-1st 6682  df-2nd 6683  df-symdif 27988  df-txp 28023  df-image 28033
This theorem is referenced by:  imageval  28100
  Copyright terms: Public domain W3C validator