Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brrangeg Structured version   Unicode version

Theorem brrangeg 25786
Description: Closed form of brrange 25784. (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 4218 . . 3  |-  ( a  =  A  ->  (
aRange b  <->  ARange b
) )
2 rneq 5098 . . . 4  |-  ( a  =  A  ->  ran  a  =  ran  A )
32eqeq2d 2449 . . 3  |-  ( a  =  A  ->  (
b  =  ran  a  <->  b  =  ran  A ) )
41, 3bibi12d 314 . 2  |-  ( a  =  A  ->  (
( aRange b  <->  b  =  ran  a )  <->  ( ARange b 
<->  b  =  ran  A
) ) )
5 breq2 4219 . . 3  |-  ( b  =  B  ->  ( ARange b  <->  ARange B ) )
6 eqeq1 2444 . . 3  |-  ( b  =  B  ->  (
b  =  ran  A  <->  B  =  ran  A ) )
75, 6bibi12d 314 . 2  |-  ( b  =  B  ->  (
( ARange b  <->  b  =  ran  A )  <->  ( ARange B  <-> 
B  =  ran  A
) ) )
8 vex 2961 . . 3  |-  a  e. 
_V
9 vex 2961 . . 3  |-  b  e. 
_V
108, 9brrange 25784 . 2  |-  ( aRange b  <->  b  =  ran  a )
114, 7, 10vtocl2g 3017 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ARange B  <->  B  =  ran  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4215   ran crn 4882  Rangecrange 25693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-eprel 4497  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-1st 6352  df-2nd 6353  df-symdif 25668  df-txp 25703  df-image 25713  df-range 25717
  Copyright terms: Public domain W3C validator