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Theorem brimageg 24466
Description: Closed form of brimage 24465. (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 4026 . . 3  |-  ( x  =  A  ->  (
xImage R y  <->  AImage R y ) )
2 imaeq2 5008 . . . 4  |-  ( x  =  A  ->  ( R " x )  =  ( R " A
) )
32eqeq2d 2294 . . 3  |-  ( x  =  A  ->  (
y  =  ( R
" x )  <->  y  =  ( R " A ) ) )
41, 3bibi12d 312 . 2  |-  ( x  =  A  ->  (
( xImage R y  <-> 
y  =  ( R
" x ) )  <-> 
( AImage R y  <-> 
y  =  ( R
" A ) ) ) )
5 breq2 4027 . . 3  |-  ( y  =  B  ->  ( AImage R y  <->  AImage R B ) )
6 eqeq1 2289 . . 3  |-  ( y  =  B  ->  (
y  =  ( R
" A )  <->  B  =  ( R " A ) ) )
75, 6bibi12d 312 . 2  |-  ( y  =  B  ->  (
( AImage R y  <-> 
y  =  ( R
" A ) )  <-> 
( AImage R B  <-> 
B  =  ( R
" A ) ) ) )
8 vex 2791 . . 3  |-  x  e. 
_V
9 vex 2791 . . 3  |-  y  e. 
_V
108, 9brimage 24465 . 2  |-  ( xImage
R y  <->  y  =  ( R " x ) )
114, 7, 10vtocl2g 2847 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( AImage R B  <-> 
B  =  ( R
" A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   "cima 4692  Imagecimage 24383
This theorem is referenced by:  fnimage  24468  funpartfv  24483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-image 24405
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