Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brdomain Structured version   Unicode version

Theorem brdomain 25779
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 25772 . 2  |-  ( AImage ( 1st  |`  ( _V  X.  _V ) ) B  <->  B  =  (
( 1st  |`  ( _V 
X.  _V ) ) " A ) )
4 df-domain 25712 . . 3  |- Domain  = Image ( 1st  |`  ( _V  X.  _V ) )
54breqi 4219 . 2  |-  ( ADomain
B  <->  AImage ( 1st  |`  ( _V  X.  _V ) ) B )
6 dfdm5 25401 . . 3  |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A
)
76eqeq2i 2447 . 2  |-  ( B  =  dom  A  <->  B  =  ( ( 1st  |`  ( _V  X.  _V ) )
" A ) )
83, 5, 73bitr4i 270 1  |-  ( ADomain
B  <->  B  =  dom  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2957   class class class wbr 4213    X. cxp 4877   dom cdm 4879    |` cres 4881   "cima 4882   1stc1st 6348  Imagecimage 25685  Domaincdomain 25688
This theorem is referenced by:  brdomaing  25781  dfrdg4  25796  tfrqfree  25797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-eprel 4495  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fo 5461  df-fv 5463  df-1st 6350  df-2nd 6351  df-symdif 25664  df-txp 25699  df-image 25709  df-domain 25712
  Copyright terms: Public domain W3C validator