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Theorem imageval 29157
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 29155 . . 3  |-  Fun Image R
2 funrel 5603 . . 3  |-  ( Fun Image R  ->  Rel Image R )
31, 2ax-mp 5 . 2  |-  Rel Image R
4 mptrel 28775 . 2  |-  Rel  (
x  e.  _V  |->  ( R " x ) )
5 vex 3116 . . . . 5  |-  y  e. 
_V
6 vex 3116 . . . . 5  |-  z  e. 
_V
75, 6breldm 5205 . . . 4  |-  ( yImage
R z  ->  y  e.  dom Image R )
8 fnimage 29156 . . . . 5  |- Image R  Fn  { x  |  ( R
" x )  e. 
_V }
9 fndm 5678 . . . . 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 2565 . . 3  |-  ( yImage
R z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
125, 6breldm 5205 . . . 4  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  dom  ( x  e. 
_V  |->  ( R "
x ) ) )
13 eqid 2467 . . . . . 6  |-  ( x  e.  _V  |->  ( R
" x ) )  =  ( x  e. 
_V  |->  ( R "
x ) )
1413dmmpt 5500 . . . . 5  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  e.  _V  |  ( R
" x )  e. 
_V }
15 rabab 3131 . . . . 5  |-  { x  e.  _V  |  ( R
" x )  e. 
_V }  =  {
x  |  ( R
" x )  e. 
_V }
1614, 15eqtri 2496 . . . 4  |-  dom  (
x  e.  _V  |->  ( R " x ) )  =  { x  |  ( R "
x )  e.  _V }
1712, 16syl6eleq 2565 . . 3  |-  ( y ( x  e.  _V  |->  ( R " x ) ) z  ->  y  e.  { x  |  ( R " x )  e.  _V } )
18 imaeq2 5331 . . . . . 6  |-  ( x  =  y  ->  ( R " x )  =  ( R " y
) )
1918eleq1d 2536 . . . . 5  |-  ( x  =  y  ->  (
( R " x
)  e.  _V  <->  ( R " y )  e.  _V ) )
205, 19elab 3250 . . . 4  |-  ( y  e.  { x  |  ( R " x
)  e.  _V }  <->  ( R " y )  e.  _V )
215, 6brimage 29153 . . . . 5  |-  ( yImage
R z  <->  z  =  ( R " y ) )
22 eqcom 2476 . . . . . 6  |-  ( z  =  ( R "
y )  <->  ( R " y )  =  z )
2318, 13fvmptg 5946 . . . . . . . . 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 2469 . . . . . . 7  |-  ( ( R " y )  e.  _V  ->  (
( ( x  e. 
_V  |->  ( R "
x ) ) `  y )  =  z  <-> 
( R " y
)  =  z ) )
26 funmpt 5622 . . . . . . . . 9  |-  Fun  (
x  e.  _V  |->  ( R " x ) )
27 df-fn 5589 . . . . . . . . 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 918 . . . . . . . 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 5906 . . . . . . . 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 5096 1  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   {cab 2452   {crab 2818   _Vcvv 3113   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   "cima 5002   Rel wrel 5004   Fun wfun 5580    Fn wfn 5581   ` cfv 5586  Imagecimage 29066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-eprel 4791  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-1st 6781  df-2nd 6782  df-symdif 29045  df-txp 29080  df-image 29090
This theorem is referenced by:  fvimage  29158
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