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Theorem brcart 29145
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 4704 . 2  |-  <. A ,  B >.  e.  _V
2 brcart.3 . 2  |-  C  e. 
_V
3 df-cart 29077 . 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 5038 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 5022 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 913 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 3anass 972 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  _E  A  /\  z  _E  B ) ) )
104epelc 4786 . . . . . . 7  |-  ( y  _E  A  <->  y  e.  A )
115epelc 4786 . . . . . . 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 1640 . . 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 3109 . . . 4  |-  x  e. 
_V
1716, 4, 5brpprod3b 29100 . . 3  |-  ( xpprod (  _E  ,  _E  ) <. A ,  B >.  <->  E. y E. z ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B ) )
18 elxp 5009 . . 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 29109 1  |-  ( <. A ,  B >.Cart C  <-> 
C  =  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   _Vcvv 3106   <.cop 4026   class class class wbr 4440    _E cep 4782    X. cxp 4990  pprodcpprod 29043  Cartccart 29053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-eprel 4784  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-1st 6774  df-2nd 6775  df-symdif 29031  df-txp 29066  df-pprod 29067  df-cart 29077
This theorem is referenced by:  brimg  29150  brrestrict  29162
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