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Theorem brrangeg 30703
Description: Closed form of brrange 30701. (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 4405 . . 3  |-  ( a  =  A  ->  (
aRange b  <->  ARange b
) )
2 rneq 5060 . . . 4  |-  ( a  =  A  ->  ran  a  =  ran  A )
32eqeq2d 2461 . . 3  |-  ( a  =  A  ->  (
b  =  ran  a  <->  b  =  ran  A ) )
41, 3bibi12d 323 . 2  |-  ( a  =  A  ->  (
( aRange b  <->  b  =  ran  a )  <->  ( ARange b 
<->  b  =  ran  A
) ) )
5 breq2 4406 . . 3  |-  ( b  =  B  ->  ( ARange b  <->  ARange B ) )
6 eqeq1 2455 . . 3  |-  ( b  =  B  ->  (
b  =  ran  A  <->  B  =  ran  A ) )
75, 6bibi12d 323 . 2  |-  ( b  =  B  ->  (
( ARange b  <->  b  =  ran  A )  <->  ( ARange B  <-> 
B  =  ran  A
) ) )
8 vex 3048 . . 3  |-  a  e. 
_V
9 vex 3048 . . 3  |-  b  e. 
_V
108, 9brrange 30701 . 2  |-  ( aRange b  <->  b  =  ran  a )
114, 7, 10vtocl2g 3111 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ARange B  <->  B  =  ran  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   class class class wbr 4402   ran crn 4835  Rangecrange 30610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-symdif 3663  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-eprel 4745  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-1st 6793  df-2nd 6794  df-txp 30620  df-image 30630  df-range 30634
This theorem is referenced by: (None)
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