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Theorem brcart 29787
Description: Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcart.1  |-  A  e. 
_V
brcart.2  |-  B  e. 
_V
brcart.3  |-  C  e. 
_V
Assertion
Ref Expression
brcart  |-  ( <. A ,  B >.Cart C  <-> 
C  =  ( A  X.  B ) )

Proof of Theorem brcart
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4720 . 2  |-  <. A ,  B >.  e.  _V
2 brcart.3 . 2  |-  C  e. 
_V
3 df-cart 29719 . 2  |- Cart  =  ( ( ( _V  X.  _V )  X.  _V )  \  ran  ( ( _V 
(x)  _E  )  /_\  (pprod (  _E  ,  _E  )  (x)  _V ) ) )
4 brcart.1 . . . 4  |-  A  e. 
_V
5 brcart.2 . . . 4  |-  B  e. 
_V
64, 5opelvv 5055 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 5039 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 920 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 3anass 977 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  _E  A  /\  z  _E  B ) ) )
104epelc 4802 . . . . . . 7  |-  ( y  _E  A  <->  y  e.  A )
115epelc 4802 . . . . . . 7  |-  ( z  _E  B  <->  z  e.  B )
1210, 11anbi12i 697 . . . . . 6  |-  ( ( y  _E  A  /\  z  _E  B )  <->  ( y  e.  A  /\  z  e.  B )
)
1312anbi2i 694 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  (
y  _E  A  /\  z  _E  B )
)  <->  ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
149, 13bitri 249 . . . 4  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B ) ) )
15142exbii 1669 . . 3  |-  ( E. y E. z ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
16 vex 3112 . . . 4  |-  x  e. 
_V
1716, 4, 5brpprod3b 29742 . . 3  |-  ( xpprod (  _E  ,  _E  ) <. A ,  B >.  <->  E. y E. z ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B ) )
18 elxp 5025 . . 3  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
1915, 17, 183bitr4ri 278 . 2  |-  ( x  e.  ( A  X.  B )  <->  xpprod (  _E  ,  _E  ) <. A ,  B >. )
201, 2, 3, 8, 19brtxpsd3 29751 1  |-  ( <. A ,  B >.Cart C  <-> 
C  =  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   _Vcvv 3109   <.cop 4038   class class class wbr 4456    _E cep 4798    X. cxp 5006  pprodcpprod 29685  Cartccart 29695
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-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-pprod 29709  df-cart 29719
This theorem is referenced by:  brimg  29792  brrestrict  29804
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