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Theorem brimageg 27956
Description: Closed form of brimage 27955. (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 4293 . . 3  |-  ( x  =  A  ->  (
xImage R y  <->  AImage R y ) )
2 imaeq2 5163 . . . 4  |-  ( x  =  A  ->  ( R " x )  =  ( R " A
) )
32eqeq2d 2452 . . 3  |-  ( x  =  A  ->  (
y  =  ( R
" x )  <->  y  =  ( R " A ) ) )
41, 3bibi12d 321 . 2  |-  ( x  =  A  ->  (
( xImage R y  <-> 
y  =  ( R
" x ) )  <-> 
( AImage R y  <-> 
y  =  ( R
" A ) ) ) )
5 breq2 4294 . . 3  |-  ( y  =  B  ->  ( AImage R y  <->  AImage R B ) )
6 eqeq1 2447 . . 3  |-  ( y  =  B  ->  (
y  =  ( R
" A )  <->  B  =  ( R " A ) ) )
75, 6bibi12d 321 . 2  |-  ( y  =  B  ->  (
( AImage R y  <-> 
y  =  ( R
" A ) )  <-> 
( AImage R B  <-> 
B  =  ( R
" A ) ) ) )
8 vex 2973 . . 3  |-  x  e. 
_V
9 vex 2973 . . 3  |-  y  e. 
_V
108, 9brimage 27955 . 2  |-  ( xImage
R y  <->  y  =  ( R " x ) )
114, 7, 10vtocl2g 3032 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( AImage R B  <-> 
B  =  ( R
" A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4290   "cima 4841  Imagecimage 27868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-eprel 4630  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fo 5422  df-fv 5424  df-1st 6575  df-2nd 6576  df-symdif 27847  df-txp 27882  df-image 27892
This theorem is referenced by:  fnimage  27958
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