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Theorem brimage 29139
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 29076 . 2  |- Image R  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  o.  `' R )  (x)  _V ) ) )
4 brxp 5022 . . 3  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
51, 2, 4mpbir2an 913 . 2  |-  A ( _V  X.  _V ) B
6 vex 3109 . . . . 5  |-  x  e. 
_V
7 vex 3109 . . . . 5  |-  y  e. 
_V
86, 7brcnv 5176 . . . 4  |-  ( x `' R y  <->  y R x )
98rexbii 2958 . . 3  |-  ( E. y  e.  A  x `' R y  <->  E. y  e.  A  y R x )
106, 1coep 28743 . . 3  |-  ( x (  _E  o.  `' R ) A  <->  E. y  e.  A  x `' R y )
116elima 5333 . . 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 29109 1  |-  ( AImage
R B  <->  B  =  ( R " A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374    e. wcel 1762   E.wrex 2808   _Vcvv 3106   class class class wbr 4440    _E cep 4782    X. cxp 4990   `'ccnv 4991   "cima 4995    o. ccom 4996  Imagecimage 29052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-eprel 4784  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-1st 6774  df-2nd 6775  df-symdif 29031  df-txp 29066  df-image 29076
This theorem is referenced by:  brimageg  29140  funimage  29141  fnimage  29142  imageval  29143  brdomain  29146  brrange  29147  brimg  29150  funpartlem  29155  imagesset  29166
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