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Theorem brcart 30705
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 4637 . 2  |-  <. A ,  B >.  e.  _V
2 brcart.3 . 2  |-  C  e. 
_V
3 df-cart 30637 . 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 4859 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4843 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 931 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 3anass 990 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  _E  A  /\  z  _E  B ) ) )
104epelc 4725 . . . . . . 7  |-  ( y  _E  A  <->  y  e.  A )
115epelc 4725 . . . . . . 7  |-  ( z  _E  B  <->  z  e.  B )
1210, 11anbi12i 708 . . . . . 6  |-  ( ( y  _E  A  /\  z  _E  B )  <->  ( y  e.  A  /\  z  e.  B )
)
1312anbi2i 705 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  (
y  _E  A  /\  z  _E  B )
)  <->  ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
149, 13bitri 257 . . . 4  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B ) ) )
15142exbii 1723 . . 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 3016 . . . 4  |-  x  e. 
_V
1716, 4, 5brpprod3b 30660 . . 3  |-  ( xpprod (  _E  ,  _E  ) <. A ,  B >.  <->  E. y E. z ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B ) )
18 elxp 4829 . . 3  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
1915, 17, 183bitr4ri 286 . 2  |-  ( x  e.  ( A  X.  B )  <->  xpprod (  _E  ,  _E  ) <. A ,  B >. )
201, 2, 3, 8, 19brtxpsd3 30669 1  |-  ( <. A ,  B >.Cart C  <-> 
C  =  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    /\ w3a 986    = wceq 1448   E.wex 1667    e. wcel 1891   _Vcvv 3013   <.cop 3942   class class class wbr 4374    _E cep 4721    X. cxp 4810  pprodcpprod 30603  Cartccart 30613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-symdif 3631  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-mpt 4435  df-eprel 4723  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-fo 5567  df-fv 5569  df-1st 6781  df-2nd 6782  df-txp 30626  df-pprod 30627  df-cart 30637
This theorem is referenced by:  brimg  30710  brrestrict  30722
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