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Theorem brrange 29815
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 29807 . 2  |-  ( AImage ( 2nd  |`  ( _V  X.  _V ) ) B  <->  B  =  (
( 2nd  |`  ( _V 
X.  _V ) ) " A ) )
4 df-range 29748 . . 3  |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
54breqi 4445 . 2  |-  ( ARange
B  <->  AImage ( 2nd  |`  ( _V  X.  _V ) ) B )
6 dfrn5 29450 . . 3  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)
76eqeq2i 2472 . 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 1398    e. wcel 1823   _Vcvv 3106   class class class wbr 4439    X. cxp 4986   ran crn 4989    |` cres 4990   "cima 4991   2ndc2nd 6772  Imagecimage 29720  Rangecrange 29724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-symdif 3715  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-eprel 4780  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773  df-2nd 6774  df-txp 29734  df-image 29744  df-range 29748
This theorem is referenced by:  brrangeg  29817  brrestrict  29830
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