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Theorem brdomaing 29553
Description: Closed form of brdomain 29551. (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 4436 . . 3  |-  ( a  =  A  ->  (
aDomain b  <->  ADomain b ) )
2 dmeq 5189 . . . 4  |-  ( a  =  A  ->  dom  a  =  dom  A )
32eqeq2d 2455 . . 3  |-  ( a  =  A  ->  (
b  =  dom  a  <->  b  =  dom  A ) )
41, 3bibi12d 321 . 2  |-  ( a  =  A  ->  (
( aDomain b  <->  b  =  dom  a )  <->  ( ADomain b 
<->  b  =  dom  A
) ) )
5 breq2 4437 . . 3  |-  ( b  =  B  ->  ( ADomain b  <->  ADomain B ) )
6 eqeq1 2445 . . 3  |-  ( b  =  B  ->  (
b  =  dom  A  <->  B  =  dom  A ) )
75, 6bibi12d 321 . 2  |-  ( b  =  B  ->  (
( ADomain b  <->  b  =  dom  A )  <->  ( ADomain B  <-> 
B  =  dom  A
) ) )
8 vex 3096 . . 3  |-  a  e. 
_V
9 vex 3096 . . 3  |-  b  e. 
_V
108, 9brdomain 29551 . 2  |-  ( aDomain b  <->  b  =  dom  a )
114, 7, 10vtocl2g 3155 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ADomain B  <->  B  =  dom  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   class class class wbr 4433   dom cdm 4985  Domaincdomain 29460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-eprel 4777  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fo 5580  df-fv 5582  df-1st 6781  df-2nd 6782  df-symdif 29436  df-txp 29471  df-image 29481  df-domain 29484
This theorem is referenced by: (None)
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