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Theorem imageval 30268
Description: The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
imageval  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
Distinct variable group:    x, R

Proof of Theorem imageval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funimage 30266 . . 3  |-  Fun Image R
2 funrel 5586 . . 3  |-  ( Fun Image R  ->  Rel Image R )
31, 2ax-mp 5 . 2  |-  Rel Image R
4 mptrel 4950 . 2  |-  Rel  (
x  e.  _V  |->  ( R " x ) )
5 vex 3062 . . . . 5  |-  y  e. 
_V
6 vex 3062 . . . . 5  |-  z  e. 
_V
75, 6breldm 5028 . . . 4  |-  ( yImage
R z  ->  y  e.  dom Image R )
8 fnimage 30267 . . . . 5  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
9 fndm 5661 . . . . 5  |-  (Image R  Fn  { x  |  ( R " x )  e.  _V }  ->  dom Image R  =  { x  |  ( R "
x )  e.  _V } )
108, 9ax-mp 5 . . . 4  |-  dom Image R  =  { x  |  ( R " x )  e.  _V }
117, 10syl6eleq 2500 . . 3  |-  ( yImage
R z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
125, 6breldm 5028 . . . 4  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  dom  ( x  e. 
_V  |->  ( R "
x ) ) )
13 eqid 2402 . . . . . 6  |-  ( x  e.  _V  |->  ( R
" x ) )  =  ( x  e. 
_V  |->  ( R "
x ) )
1413dmmpt 5318 . . . . 5  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  e.  _V  |  ( R
" x )  e. 
_V }
15 rabab 3077 . . . . 5  |-  { x  e.  _V  |  ( R
" x )  e. 
_V }  =  {
x  |  ( R
" x )  e. 
_V }
1614, 15eqtri 2431 . . . 4  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  |  ( R "
x )  e.  _V }
1712, 16syl6eleq 2500 . . 3  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
18 imaeq2 5153 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2471 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
205, 19elab 3196 . . . 4  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  <->  ( R " y )  e.  _V )
215, 6brimage 30264 . . . . 5  |-  ( yImage
R z  <->  z  =  ( R " y ) )
22 eqcom 2411 . . . . . 6  |-  ( z  =  ( R "
y )  <->  ( R " y )  =  z )
2318, 13fvmptg 5930 . . . . . . . . 9  |-  ( ( y  e.  _V  /\  ( R " y )  e.  _V )  -> 
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  ( R " y ) )
245, 23mpan 668 . . . . . . . 8  |-  ( ( R " y )  e.  _V  ->  (
( x  e.  _V  |->  ( R " x ) ) `  y )  =  ( R "
y ) )
2524eqeq1d 2404 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  z  <-> 
( R " y
)  =  z ) )
26 funmpt 5605 . . . . . . . . 9  |-  Fun  (
x  e.  _V  |->  ( R " x ) )
27 df-fn 5572 . . . . . . . . 9  |-  ( ( x  e.  _V  |->  ( R " x ) )  Fn  { x  |  ( R "
x )  e.  _V } 
<->  ( Fun  ( x  e.  _V  |->  ( R
" x ) )  /\  dom  ( x  e.  _V  |->  ( R
" x ) )  =  { x  |  ( R " x
)  e.  _V }
) )
2826, 16, 27mpbir2an 921 . . . . . . . 8  |-  ( x  e.  _V  |->  ( R
" x ) )  Fn  { x  |  ( R " x
)  e.  _V }
2920biimpri 206 . . . . . . . 8  |-  ( ( R " y )  e.  _V  ->  y  e.  { x  |  ( R " x )  e.  _V } )
30 fnbrfvb 5889 . . . . . . . 8  |-  ( ( ( x  e.  _V  |->  ( R " x ) )  Fn  { x  |  ( R "
x )  e.  _V }  /\  y  e.  {
x  |  ( R
" x )  e. 
_V } )  -> 
( ( ( x  e.  _V  |->  ( R
" x ) ) `
 y )  =  z  <->  y ( x  e.  _V  |->  ( R
" x ) ) z ) )
3128, 29, 30sylancr 661 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  z  <-> 
y ( x  e. 
_V  |->  ( R "
x ) ) z ) )
3225, 31bitr3d 255 . . . . . 6  |-  ( ( R " y )  e.  _V  ->  (
( R " y
)  =  z  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3322, 32syl5bb 257 . . . . 5  |-  ( ( R " y )  e.  _V  ->  (
z  =  ( R
" y )  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3421, 33syl5bb 257 . . . 4  |-  ( ( R " y )  e.  _V  ->  (
yImage R z  <->  y (
x  e.  _V  |->  ( R " x ) ) z ) )
3520, 34sylbi 195 . . 3  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  ->  ( yImage R z  <-> 
y ( x  e. 
_V  |->  ( R "
x ) ) z ) )
3611, 17, 35pm5.21nii 351 . 2  |-  ( yImage
R z  <->  y (
x  e.  _V  |->  ( R " x ) ) z )
373, 4, 36eqbrriv 4919 1  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405    e. wcel 1842   {cab 2387   {crab 2758   _Vcvv 3059   class class class wbr 4395    |-> cmpt 4453   dom cdm 4823   "cima 4826   Rel wrel 4828   Fun wfun 5563    Fn wfn 5564   ` cfv 5569  Imagecimage 30177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-symdif 3670  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-eprel 4734  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577  df-1st 6784  df-2nd 6785  df-txp 30191  df-image 30201
This theorem is referenced by:  fvimage  30269
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