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Theorem brimage 28094
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 28031 . 2  |- Image R  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  o.  `' R )  (x)  _V ) ) )
4 brxp 4971 . . 3  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
51, 2, 4mpbir2an 911 . 2  |-  A ( _V  X.  _V ) B
6 vex 3074 . . . . 5  |-  x  e. 
_V
7 vex 3074 . . . . 5  |-  y  e. 
_V
86, 7brcnv 5123 . . . 4  |-  ( x `' R y  <->  y R x )
98rexbii 2859 . . 3  |-  ( E. y  e.  A  x `' R y  <->  E. y  e.  A  y R x )
106, 1coep 27698 . . 3  |-  ( x (  _E  o.  `' R ) A  <->  E. y  e.  A  x `' R y )
116elima 5275 . . 3  |-  ( x  e.  ( R " A )  <->  E. y  e.  A  y R x )
129, 10, 113bitr4ri 278 . 2  |-  ( x  e.  ( R " A )  <->  x (  _E  o.  `' R ) A )
131, 2, 3, 5, 12brtxpsd3 28064 1  |-  ( AImage
R B  <->  B  =  ( R " A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3071   class class class wbr 4393    _E cep 4731    X. cxp 4939   `'ccnv 4940   "cima 4944    o. ccom 4945  Imagecimage 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-eprel 4733  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fo 5525  df-fv 5527  df-1st 6680  df-2nd 6681  df-symdif 27986  df-txp 28021  df-image 28031
This theorem is referenced by:  brimageg  28095  funimage  28096  fnimage  28097  imageval  28098  brdomain  28101  brrange  28102  brimg  28105  funpartlem  28110  imagesset  28121
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