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Theorem brrangeg 30261
Description: Closed form of brrange 30259. (Contributed by Scott Fenton, 3-May-2014.)
Assertion
Ref Expression
brrangeg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ARange B  <->  B  =  ran  A ) )

Proof of Theorem brrangeg
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4397 . . 3  |-  ( a  =  A  ->  (
aRange b  <->  ARange b
) )
2 rneq 5048 . . . 4  |-  ( a  =  A  ->  ran  a  =  ran  A )
32eqeq2d 2416 . . 3  |-  ( a  =  A  ->  (
b  =  ran  a  <->  b  =  ran  A ) )
41, 3bibi12d 319 . 2  |-  ( a  =  A  ->  (
( aRange b  <->  b  =  ran  a )  <->  ( ARange b 
<->  b  =  ran  A
) ) )
5 breq2 4398 . . 3  |-  ( b  =  B  ->  ( ARange b  <->  ARange B ) )
6 eqeq1 2406 . . 3  |-  ( b  =  B  ->  (
b  =  ran  A  <->  B  =  ran  A ) )
75, 6bibi12d 319 . 2  |-  ( b  =  B  ->  (
( ARange b  <->  b  =  ran  A )  <->  ( ARange B  <-> 
B  =  ran  A
) ) )
8 vex 3061 . . 3  |-  a  e. 
_V
9 vex 3061 . . 3  |-  b  e. 
_V
108, 9brrange 30259 . 2  |-  ( aRange b  <->  b  =  ran  a )
114, 7, 10vtocl2g 3120 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ARange B  <->  B  =  ran  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   class class class wbr 4394   ran crn 4823  Rangecrange 30168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-symdif 3669  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-eprel 4733  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-1st 6783  df-2nd 6784  df-txp 30178  df-image 30188  df-range 30192
This theorem is referenced by: (None)
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