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Theorem brimageg 29739
Description: Closed form of brimage 29738. (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 4459 . . 3  |-  ( x  =  A  ->  (
xImage R y  <->  AImage R y ) )
2 imaeq2 5343 . . . 4  |-  ( x  =  A  ->  ( R " x )  =  ( R " A
) )
32eqeq2d 2471 . . 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 4460 . . 3  |-  ( y  =  B  ->  ( AImage R y  <->  AImage R B ) )
6 eqeq1 2461 . . 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 3112 . . 3  |-  x  e. 
_V
9 vex 3112 . . 3  |-  y  e. 
_V
108, 9brimage 29738 . 2  |-  ( xImage
R y  <->  y  =  ( R " x ) )
114, 7, 10vtocl2g 3171 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 1395    e. wcel 1819   class class class wbr 4456   "cima 5011  Imagecimage 29651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-symdif 3725  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-eprel 4800  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-1st 6799  df-2nd 6800  df-txp 29665  df-image 29675
This theorem is referenced by:  fnimage  29741
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