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Theorem fvimage 28093
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 3074 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 imaeq2 5260 . . 3  |-  ( x  =  A  ->  ( R " x )  =  ( R " A
) )
3 imageval 28092 . . 3  |- Image R  =  ( x  e.  _V  |->  ( R " x ) )
42, 3fvmptg 5868 . 2  |-  ( ( A  e.  _V  /\  ( R " A )  e.  W )  -> 
(Image R `  A
)  =  ( R
" A ) )
51, 4sylan 471 1  |-  ( ( A  e.  V  /\  ( R " A )  e.  W )  -> 
(Image R `  A
)  =  ( R
" A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3065   "cima 4938   ` cfv 5513  Imagecimage 28001
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 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-eprel 4727  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fo 5519  df-fv 5521  df-1st 6674  df-2nd 6675  df-symdif 27980  df-txp 28015  df-image 28025
This theorem is referenced by: (None)
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