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Theorem brrange 28104
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 28096 . 2  |-  ( AImage ( 2nd  |`  ( _V  X.  _V ) ) B  <->  B  =  (
( 2nd  |`  ( _V 
X.  _V ) ) " A ) )
4 df-range 28037 . . 3  |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
54breqi 4401 . 2  |-  ( ARange
B  <->  AImage ( 2nd  |`  ( _V  X.  _V ) ) B )
6 dfrn5 27727 . . 3  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)
76eqeq2i 2470 . 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 1370    e. wcel 1758   _Vcvv 3072   class class class wbr 4395    X. cxp 4941   ran crn 4944    |` cres 4945   "cima 4946   2ndc2nd 6681  Imagecimage 28009  Rangecrange 28013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-eprel 4735  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529  df-1st 6682  df-2nd 6683  df-symdif 27988  df-txp 28023  df-image 28033  df-range 28037
This theorem is referenced by:  brrangeg  28106  brrestrict  28119
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