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Theorem imageval 27976
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 27974 . . 3  |-  Fun Image R
2 funrel 5450 . . 3  |-  ( Fun Image R  ->  Rel Image R )
31, 2ax-mp 5 . 2  |-  Rel Image R
4 mptrel 27594 . 2  |-  Rel  (
x  e.  _V  |->  ( R " x ) )
5 vex 2990 . . . . 5  |-  y  e. 
_V
6 vex 2990 . . . . 5  |-  z  e. 
_V
75, 6breldm 5059 . . . 4  |-  ( yImage
R z  ->  y  e.  dom Image R )
8 fnimage 27975 . . . . 5  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
9 fndm 5525 . . . . 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 2533 . . 3  |-  ( yImage
R z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
125, 6breldm 5059 . . . 4  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  dom  ( x  e. 
_V  |->  ( R "
x ) ) )
13 eqid 2443 . . . . . 6  |-  ( x  e.  _V  |->  ( R
" x ) )  =  ( x  e. 
_V  |->  ( R "
x ) )
1413dmmpt 5348 . . . . 5  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  e.  _V  |  ( R
" x )  e. 
_V }
15 rabab 3005 . . . . 5  |-  { x  e.  _V  |  ( R
" x )  e. 
_V }  =  {
x  |  ( R
" x )  e. 
_V }
1614, 15eqtri 2463 . . . 4  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  |  ( R "
x )  e.  _V }
1712, 16syl6eleq 2533 . . 3  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
18 imaeq2 5180 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2509 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
205, 19elab 3121 . . . 4  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  <->  ( R " y )  e.  _V )
215, 6brimage 27972 . . . . 5  |-  ( yImage
R z  <->  z  =  ( R " y ) )
22 eqcom 2445 . . . . . 6  |-  ( z  =  ( R "
y )  <->  ( R " y )  =  z )
2318, 13fvmptg 5787 . . . . . . . . 9  |-  ( ( y  e.  _V  /\  ( R " y )  e.  _V )  -> 
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  ( R " y ) )
245, 23mpan 670 . . . . . . . 8  |-  ( ( R " y )  e.  _V  ->  (
( x  e.  _V  |->  ( R " x ) ) `  y )  =  ( R "
y ) )
2524eqeq1d 2451 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  z  <-> 
( R " y
)  =  z ) )
26 funmpt 5469 . . . . . . . . 9  |-  Fun  (
x  e.  _V  |->  ( R " x ) )
27 df-fn 5436 . . . . . . . . 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 911 . . . . . . . 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 5747 . . . . . . . 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 663 . . . . . . 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 353 . 2  |-  ( yImage
R z  <->  y (
x  e.  _V  |->  ( R " x ) ) z )
373, 4, 36eqbrriv 4950 1  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756   {cab 2429   {crab 2734   _Vcvv 2987   class class class wbr 4307    e. cmpt 4365   dom cdm 4855   "cima 4858   Rel wrel 4860   Fun wfun 5427    Fn wfn 5428   ` cfv 5433  Imagecimage 27885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-eprel 4647  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-fo 5439  df-fv 5441  df-1st 6592  df-2nd 6593  df-symdif 27864  df-txp 27899  df-image 27909
This theorem is referenced by:  fvimage  27977
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