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Theorem brrange 25781
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 25773 . 2  |-  ( AImage ( 2nd  |`  ( _V  X.  _V ) ) B  <->  B  =  (
( 2nd  |`  ( _V 
X.  _V ) ) " A ) )
4 df-range 25714 . . 3  |- Range  = Image ( 2nd  |`  ( _V  X.  _V ) )
54breqi 4220 . 2  |-  ( ARange
B  <->  AImage ( 2nd  |`  ( _V  X.  _V ) ) B )
6 dfrn5 25403 . . 3  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)
76eqeq2i 2448 . 2  |-  ( B  =  ran  A  <->  B  =  ( ( 2nd  |`  ( _V  X.  _V ) )
" A ) )
83, 5, 73bitr4i 270 1  |-  ( ARange
B  <->  B  =  ran  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958   class class class wbr 4214    X. cxp 4878   ran crn 4881    |` cres 4882   "cima 4883   2ndc2nd 6350  Imagecimage 25686  Rangecrange 25690
This theorem is referenced by:  brrangeg  25783  brrestrict  25796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-eprel 4496  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-1st 6351  df-2nd 6352  df-symdif 25665  df-txp 25700  df-image 25710  df-range 25714
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