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Theorem brrange 24544
Description: The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brdomain.1  |-  A  e. 
_V
brdomain.2  |-  B  e. 
_V
Assertion
Ref Expression
brrange  |-  ( ARange
B  <->  B  =  ran  A )

Proof of Theorem brrange
StepHypRef Expression
1 brdomain.1 . . 3  |-  A  e. 
_V
2 brdomain.2 . . 3  |-  B  e. 
_V
31, 2brimage 24536 . 2  |-  ( AImage ( 2nd  |`  ( _V  X.  _V ) ) B  <->  B  =  (
( 2nd  |`  ( _V 
X.  _V ) ) " A ) )
4 df-range 24480 . . 3  |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
54breqi 4045 . 2  |-  ( ARange
B  <->  AImage ( 2nd  |`  ( _V  X.  _V ) ) B )
6 dfrn5 24204 . . 3  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)
76eqeq2i 2306 . 2  |-  ( B  =  ran  A  <->  B  =  ( ( 2nd  |`  ( _V  X.  _V ) )
" A ) )
83, 5, 73bitr4i 268 1  |-  ( ARange
B  <->  B  =  ran  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    X. cxp 4703   ran crn 4706    |` cres 4707   "cima 4708   2ndc2nd 6137  Imagecimage 24454  Rangecrange 24458
This theorem is referenced by:  brrangeg  24546  brrestrict  24559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-image 24476  df-range 24480
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