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Theorem brimage 25490
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 25430 . 2  |- Image R  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  o.  `' R )  (x)  _V ) ) )
4 brxp 4850 . . 3  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
51, 2, 4mpbir2an 887 . 2  |-  A ( _V  X.  _V ) B
6 vex 2903 . . . . 5  |-  x  e. 
_V
7 vex 2903 . . . . 5  |-  y  e. 
_V
86, 7brcnv 4996 . . . 4  |-  ( x `' R y  <->  y R x )
98rexbii 2675 . . 3  |-  ( E. y  e.  A  x `' R y  <->  E. y  e.  A  y R x )
106, 1coep 25133 . . 3  |-  ( x (  _E  o.  `' R ) A  <->  E. y  e.  A  x `' R y )
116elima 5149 . . 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 25461 1  |-  ( AImage
R B  <->  B  =  ( R " A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   E.wrex 2651   _Vcvv 2900   class class class wbr 4154    _E cep 4434    X. cxp 4817   `'ccnv 4818   "cima 4822    o. ccom 4823  Imagecimage 25408
This theorem is referenced by:  brimageg  25491  funimage  25492  fnimage  25493  imageval  25494  brdomain  25497  brrange  25498  brimg  25501  funpartlem  25506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-eprel 4436  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-1st 6289  df-2nd 6290  df-symdif 25387  df-txp 25420  df-image 25430
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