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Theorem brdomain 29788
Description: The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brdomain.1  |-  A  e. 
_V
brdomain.2  |-  B  e. 
_V
Assertion
Ref Expression
brdomain  |-  ( ADomain
B  <->  B  =  dom  A )

Proof of Theorem brdomain
StepHypRef Expression
1 brdomain.1 . . 3  |-  A  e. 
_V
2 brdomain.2 . . 3  |-  B  e. 
_V
31, 2brimage 29781 . 2  |-  ( AImage ( 1st  |`  ( _V  X.  _V ) ) B  <->  B  =  (
( 1st  |`  ( _V 
X.  _V ) ) " A ) )
4 df-domain 29721 . . 3  |- Domain  = Image ( 1st  |`  ( _V  X.  _V ) )
54breqi 4462 . 2  |-  ( ADomain
B  <->  AImage ( 1st  |`  ( _V  X.  _V ) ) B )
6 dfdm5 29423 . . 3  |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A
)
76eqeq2i 2475 . 2  |-  ( B  =  dom  A  <->  B  =  ( ( 1st  |`  ( _V  X.  _V ) )
" A ) )
83, 5, 73bitr4i 277 1  |-  ( ADomain
B  <->  B  =  dom  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    e. wcel 1819   _Vcvv 3109   class class class wbr 4456    X. cxp 5006   dom cdm 5008    |` cres 5010   "cima 5011   1stc1st 6797  Imagecimage 29694  Domaincdomain 29697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-symdif 3725  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-eprel 4800  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-1st 6799  df-2nd 6800  df-txp 29708  df-image 29718  df-domain 29721
This theorem is referenced by:  brdomaing  29790  dfrdg4  29805  tfrqfree  29806
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