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Theorem brimageg 25774
Description: Closed form of brimage 25773. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
brimageg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( AImage R B  <-> 
B  =  ( R
" A ) ) )

Proof of Theorem brimageg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4217 . . 3  |-  ( x  =  A  ->  (
xImage R y  <->  AImage R y ) )
2 imaeq2 5201 . . . 4  |-  ( x  =  A  ->  ( R " x )  =  ( R " A
) )
32eqeq2d 2449 . . 3  |-  ( x  =  A  ->  (
y  =  ( R
" x )  <->  y  =  ( R " A ) ) )
41, 3bibi12d 314 . 2  |-  ( x  =  A  ->  (
( xImage R y  <-> 
y  =  ( R
" x ) )  <-> 
( AImage R y  <-> 
y  =  ( R
" A ) ) ) )
5 breq2 4218 . . 3  |-  ( y  =  B  ->  ( AImage R y  <->  AImage R B ) )
6 eqeq1 2444 . . 3  |-  ( y  =  B  ->  (
y  =  ( R
" A )  <->  B  =  ( R " A ) ) )
75, 6bibi12d 314 . 2  |-  ( y  =  B  ->  (
( AImage R y  <-> 
y  =  ( R
" A ) )  <-> 
( AImage R B  <-> 
B  =  ( R
" A ) ) ) )
8 vex 2961 . . 3  |-  x  e. 
_V
9 vex 2961 . . 3  |-  y  e. 
_V
108, 9brimage 25773 . 2  |-  ( xImage
R y  <->  y  =  ( R " x ) )
114, 7, 10vtocl2g 3017 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( AImage R B  <-> 
B  =  ( R
" A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4214   "cima 4883  Imagecimage 25686
This theorem is referenced by:  fnimage  25776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-eprel 4496  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-1st 6351  df-2nd 6352  df-symdif 25665  df-txp 25700  df-image 25710
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