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

Theorem fvimage 25776
Description: The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvimage  |-  ( ( A  e.  V  /\  ( R " A )  e.  W )  -> 
(Image R `  A
)  =  ( R
" A ) )

Proof of Theorem fvimage
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2964 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 imaeq2 5199 . . 3  |-  ( x  =  A  ->  ( R " x )  =  ( R " A
) )
3 imageval 25775 . . 3  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
42, 3fvmptg 5804 . 2  |-  ( ( A  e.  _V  /\  ( R " A )  e.  W )  -> 
(Image R `  A
)  =  ( R
" A ) )
51, 4sylan 458 1  |-  ( ( A  e.  V  /\  ( R " A )  e.  W )  -> 
(Image R `  A
)  =  ( R
" A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   "cima 4881   ` cfv 5454  Imagecimage 25684
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-eprel 4494  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-symdif 25663  df-txp 25698  df-image 25708
  Copyright terms: Public domain W3C validator