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Theorem brimage 25764
Description: Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brimage.1  |-  A  e. 
_V
brimage.2  |-  B  e. 
_V
Assertion
Ref Expression
brimage  |-  ( AImage
R B  <->  B  =  ( R " A ) )

Proof of Theorem brimage
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brimage.1 . 2  |-  A  e. 
_V
2 brimage.2 . 2  |-  B  e. 
_V
3 df-image 25701 . 2  |- Image R  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  o.  `' R )  (x)  _V ) ) )
4 brxp 4902 . . 3  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
51, 2, 4mpbir2an 887 . 2  |-  A ( _V  X.  _V ) B
6 vex 2952 . . . . 5  |-  x  e. 
_V
7 vex 2952 . . . . 5  |-  y  e. 
_V
86, 7brcnv 5048 . . . 4  |-  ( x `' R y  <->  y R x )
98rexbii 2723 . . 3  |-  ( E. y  e.  A  x `' R y  <->  E. y  e.  A  y R x )
106, 1coep 25367 . . 3  |-  ( x (  _E  o.  `' R ) A  <->  E. y  e.  A  x `' R y )
116elima 5201 . . 3  |-  ( x  e.  ( R " A )  <->  E. y  e.  A  y R x )
129, 10, 113bitr4ri 270 . 2  |-  ( x  e.  ( R " A )  <->  x (  _E  o.  `' R ) A )
131, 2, 3, 5, 12brtxpsd3 25734 1  |-  ( AImage
R B  <->  B  =  ( R " A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   E.wrex 2699   _Vcvv 2949   class class class wbr 4205    _E cep 4485    X. cxp 4869   `'ccnv 4870   "cima 4874    o. ccom 4875  Imagecimage 25677
This theorem is referenced by:  brimageg  25765  funimage  25766  fnimage  25767  imageval  25768  brdomain  25771  brrange  25772  brimg  25775  funpartlem  25780  imagesset  25791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-eprel 4487  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fo 5453  df-fv 5455  df-1st 6342  df-2nd 6343  df-symdif 25656  df-txp 25691  df-image 25701
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