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Theorem brrangeg 25301
Description: Closed form of brrange 25299. (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 4128 . . 3  |-  ( a  =  A  ->  (
aRange b  <->  ARange b
) )
2 rneq 5007 . . . 4  |-  ( a  =  A  ->  ran  a  =  ran  A )
32eqeq2d 2377 . . 3  |-  ( a  =  A  ->  (
b  =  ran  a  <->  b  =  ran  A ) )
41, 3bibi12d 312 . 2  |-  ( a  =  A  ->  (
( aRange b  <->  b  =  ran  a )  <->  ( ARange b 
<->  b  =  ran  A
) ) )
5 breq2 4129 . . 3  |-  ( b  =  B  ->  ( ARange b  <->  ARange B ) )
6 eqeq1 2372 . . 3  |-  ( b  =  B  ->  (
b  =  ran  A  <->  B  =  ran  A ) )
75, 6bibi12d 312 . 2  |-  ( b  =  B  ->  (
( ARange b  <->  b  =  ran  A )  <->  ( ARange B  <-> 
B  =  ran  A
) ) )
8 vex 2876 . . 3  |-  a  e. 
_V
9 vex 2876 . . 3  |-  b  e. 
_V
108, 9brrange 25299 . 2  |-  ( aRange b  <->  b  =  ran  a )
114, 7, 10vtocl2g 2932 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ARange B  <->  B  =  ran  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   class class class wbr 4125   ran crn 4793  Rangecrange 25213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-eprel 4408  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-1st 6249  df-2nd 6250  df-symdif 25188  df-txp 25221  df-image 25231  df-range 25235
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