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Theorem brdomaing 24545
Description: Closed form of brdomain 24543. (Contributed by Scott Fenton, 2-May-2014.)
Assertion
Ref Expression
brdomaing  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ADomain B  <->  B  =  dom  A ) )

Proof of Theorem brdomaing
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4042 . . 3  |-  ( a  =  A  ->  (
aDomain b  <->  ADomain b ) )
2 dmeq 4895 . . . 4  |-  ( a  =  A  ->  dom  a  =  dom  A )
32eqeq2d 2307 . . 3  |-  ( a  =  A  ->  (
b  =  dom  a  <->  b  =  dom  A ) )
41, 3bibi12d 312 . 2  |-  ( a  =  A  ->  (
( aDomain b  <->  b  =  dom  a )  <->  ( ADomain b 
<->  b  =  dom  A
) ) )
5 breq2 4043 . . 3  |-  ( b  =  B  ->  ( ADomain b  <->  ADomain B ) )
6 eqeq1 2302 . . 3  |-  ( b  =  B  ->  (
b  =  dom  A  <->  B  =  dom  A ) )
75, 6bibi12d 312 . 2  |-  ( b  =  B  ->  (
( ADomain b  <->  b  =  dom  A )  <->  ( ADomain B  <-> 
B  =  dom  A
) ) )
8 vex 2804 . . 3  |-  a  e. 
_V
9 vex 2804 . . 3  |-  b  e. 
_V
108, 9brdomain 24543 . 2  |-  ( aDomain b  <->  b  =  dom  a )
114, 7, 10vtocl2g 2860 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ADomain B  <->  B  =  dom  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705  Domaincdomain 24457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-image 24476  df-domain 24479
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