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Theorem brdomaing 24474
Description: Closed form of brdomain 24472. (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 4026 . . 3  |-  ( a  =  A  ->  (
aDomain b  <->  ADomain b ) )
2 dmeq 4879 . . . 4  |-  ( a  =  A  ->  dom  a  =  dom  A )
32eqeq2d 2294 . . 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 4027 . . 3  |-  ( b  =  B  ->  ( ADomain b  <->  ADomain B ) )
6 eqeq1 2289 . . 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 2791 . . 3  |-  a  e. 
_V
9 vex 2791 . . 3  |-  b  e. 
_V
108, 9brdomain 24472 . 2  |-  ( aDomain b  <->  b  =  dom  a )
114, 7, 10vtocl2g 2847 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 1623    e. wcel 1684   class class class wbr 4023   dom cdm 4689  Domaincdomain 24386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-image 24405  df-domain 24408
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