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Theorem brrange 29552
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 29544 . 2  |-  ( AImage ( 2nd  |`  ( _V  X.  _V ) ) B  <->  B  =  (
( 2nd  |`  ( _V 
X.  _V ) ) " A ) )
4 df-range 29485 . . 3  |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
54breqi 4439 . 2  |-  ( ARange
B  <->  AImage ( 2nd  |`  ( _V  X.  _V ) ) B )
6 dfrn5 29175 . . 3  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)
76eqeq2i 2459 . 2  |-  ( B  =  ran  A  <->  B  =  ( ( 2nd  |`  ( _V  X.  _V ) )
" A ) )
83, 5, 73bitr4i 277 1  |-  ( ARange
B  <->  B  =  ran  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1381    e. wcel 1802   _Vcvv 3093   class class class wbr 4433    X. cxp 4983   ran crn 4986    |` cres 4987   "cima 4988   2ndc2nd 6780  Imagecimage 29457  Rangecrange 29461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-eprel 4777  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fo 5580  df-fv 5582  df-1st 6781  df-2nd 6782  df-symdif 29436  df-txp 29471  df-image 29481  df-range 29485
This theorem is referenced by:  brrangeg  29554  brrestrict  29567
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