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Theorem brdomaing 25499
Description: Closed form of brdomain 25497. (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 4157 . . 3  |-  ( a  =  A  ->  (
aDomain b  <->  ADomain b ) )
2 dmeq 5011 . . . 4  |-  ( a  =  A  ->  dom  a  =  dom  A )
32eqeq2d 2399 . . 3  |-  ( a  =  A  ->  (
b  =  dom  a  <->  b  =  dom  A ) )
41, 3bibi12d 313 . 2  |-  ( a  =  A  ->  (
( aDomain b  <->  b  =  dom  a )  <->  ( ADomain b 
<->  b  =  dom  A
) ) )
5 breq2 4158 . . 3  |-  ( b  =  B  ->  ( ADomain b  <->  ADomain B ) )
6 eqeq1 2394 . . 3  |-  ( b  =  B  ->  (
b  =  dom  A  <->  B  =  dom  A ) )
75, 6bibi12d 313 . 2  |-  ( b  =  B  ->  (
( ADomain b  <->  b  =  dom  A )  <->  ( ADomain B  <-> 
B  =  dom  A
) ) )
8 vex 2903 . . 3  |-  a  e. 
_V
9 vex 2903 . . 3  |-  b  e. 
_V
108, 9brdomain 25497 . 2  |-  ( aDomain b  <->  b  =  dom  a )
114, 7, 10vtocl2g 2959 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ADomain B  <->  B  =  dom  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4154   dom cdm 4819  Domaincdomain 25411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-eprel 4436  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-1st 6289  df-2nd 6290  df-symdif 25387  df-txp 25420  df-image 25430  df-domain 25433
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