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Theorem brdomaing 27985
Description: Closed form of brdomain 27983. (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 4314 . . 3  |-  ( a  =  A  ->  (
aDomain b  <->  ADomain b ) )
2 dmeq 5059 . . . 4  |-  ( a  =  A  ->  dom  a  =  dom  A )
32eqeq2d 2454 . . 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 4315 . . 3  |-  ( b  =  B  ->  ( ADomain b  <->  ADomain B ) )
6 eqeq1 2449 . . 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 2994 . . 3  |-  a  e. 
_V
9 vex 2994 . . 3  |-  b  e. 
_V
108, 9brdomain 27983 . 2  |-  ( aDomain b  <->  b  =  dom  a )
114, 7, 10vtocl2g 3053 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 1369    e. wcel 1756   class class class wbr 4311   dom cdm 4859  Domaincdomain 27892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-mpt 4371  df-eprel 4651  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-fo 5443  df-fv 5445  df-1st 6596  df-2nd 6597  df-symdif 27868  df-txp 27903  df-image 27913  df-domain 27916
This theorem is referenced by: (None)
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