Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brimage Structured version   Unicode version

Theorem brimage 30264
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 30201 . 2  |- Image R  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  o.  `' R ) 
(x)  _V ) ) )
4 brxp 4854 . . 3  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
51, 2, 4mpbir2an 921 . 2  |-  A ( _V  X.  _V ) B
6 vex 3062 . . . . 5  |-  x  e. 
_V
7 vex 3062 . . . . 5  |-  y  e. 
_V
86, 7brcnv 5006 . . . 4  |-  ( x `' R y  <->  y R x )
98rexbii 2906 . . 3  |-  ( E. y  e.  A  x `' R y  <->  E. y  e.  A  y R x )
106, 1coep 29964 . . 3  |-  ( x (  _E  o.  `' R ) A  <->  E. y  e.  A  x `' R y )
116elima 5162 . . 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 30234 1  |-  ( AImage
R B  <->  B  =  ( R " A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405    e. wcel 1842   E.wrex 2755   _Vcvv 3059   class class class wbr 4395    _E cep 4732    X. cxp 4821   `'ccnv 4822   "cima 4826    o. ccom 4827  Imagecimage 30177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-symdif 3670  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-eprel 4734  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577  df-1st 6784  df-2nd 6785  df-txp 30191  df-image 30201
This theorem is referenced by:  brimageg  30265  funimage  30266  fnimage  30267  imageval  30268  brdomain  30271  brrange  30272  brimg  30275  funpartlem  30280  imagesset  30291
  Copyright terms: Public domain W3C validator